System and method for estimating indoor temperature time series data of a building with the aid of a digital computer

ABSTRACT

A system and method to determine building thermal performance parameters through empirical testing is described. The parameters can be formulaically applied to determine fuel consumption and indoor temperatures. To generalize the approach, the term used to represent furnace rating is replaced with HVAC system rating. As total heat change is based on the building&#39;s thermal mass, heat change is relabeled as thermal mass gain (or loss). This change creates a heat balance equation that is composed of heat gain (loss) from six sources, three of which contribute to heat gain only. No modifications are required for apply the empirical tests to summer since an attic&#39;s thermal conductivity cancels out and the attic&#39;s effective window area is directly combined with the existing effective window area. Since these tests are empirically based, the tests already account for the additional heat gain associated with the elevated attic temperature and other surface temperatures.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation-in-part of U.S. patentapplication Ser. No. 14/664,742, filed Mar. 20, 2015, pending, which isa continuation of U.S. patent application Ser. No. 14/631,798, filedFeb. 25, 2015, pending, the priority dates of which are claimed and thedisclosures of which are incorporated by reference.

FIELD

This application relates in general to energy conservation and, inparticular, to a system and method for estimating indoor temperaturetime series data of a building with the aid of a digital computer.

BACKGROUND

Concern has been growing in recent times over energy consumption in theUnited States and abroad. The cost of energy has steadily risen as powerutilities try to cope with continually growing demand, increasing fuelprices, and stricter regulatory mandates. Power utilities must alsomaintain existing infrastructure, while simultaneously finding ways toadd more generation capacity to meet future needs, both of which add tothe cost of energy. Moreover, burgeoning energy consumption continues toimpact the environment and deplete natural resources.

A major portion of the rising cost of energy is borne by consumers, who,despite the need, remain poorly-equipped to identify the most costeffective ways to lower their energy consumption. No-cost behavioralchanges, such as adjusting thermostat settings and turning off unusedappliances, and low-cost physical improvements, such as switching toenergy-efficient light bulbs, may be insufficient to offset increases inmonthly utility bills, particularly as seasonal space heating and airconditioning (AC) together consume the most energy in the average home.As a result, appreciable decreases in energy consumption can usuallyonly be achieved by investing in upgrades to a building's heating orcooling envelope or “shell.” However, identifying and comparing thoseimprovements to a building's shell that will yield an acceptable returnon investment in terms of costs versus energy savings requiresdetermining building-specific parameters, especially the building'sthermal conductivity (UA^(Total)).

Heating, ventilating, and air conditioning (HVAC) energy costs are tiedto a building's thermal conductivity UA^(Total). A poorly insulated homeor a building with significant sealing problems will require more HVACusage to maintain a desired interior temperature than would acomparably-sized but well-insulated and sealed structure. Reducing HVACenergy costs is not as simple as choosing a thermostat setting thatcauses an HVAC system to run for less time or less often. Rather, HVACsystem efficiency, duration of heating or cooling seasons, differencesbetween indoor and outdoor temperatures, and other factors combined withthermal conductivity UA^(Total) can weigh into overall energyconsumption.

Conventionally, an on-site energy audit is performed to determine abuilding's thermal conductivity UA^(Total). A typical energy auditinvolves measuring the dimensions of walls, windows, doors, and otherphysical characteristics;

approximating R-values of insulation for thermal resistance; estimatinginfiltration using a blower door test; and detecting air leakage using athermal camera, after which a numerical model is run to solve forthermal conductivity. The UA^(Total) result is combined with theduration of the heating or cooling season, as applicable, over theperiod of inquiry and adjusted for HVAC system efficiency, plus anysolar (or other non-utility supplied) power savings fraction. The auditreport is often presented in the form of a checklist of correctivemeasures that may be taken to improve the building's shell and HVACsystem, and thereby lower overall energy consumption. As an involvedprocess, an energy audit can be costly, time-consuming, and invasive forbuilding owners and occupants. Further, as a numerical result derivedfrom a theoretical model, an energy audit carries an inherent potentialfor inaccuracy strongly influenced by physical mismeasurements, lack ofmeasurements, data assumptions, and so forth. As well, the degree ofimprovement and savings attributable to various possible improvements isnot necessarily quantified due to the wide range of variables.

Therefore, a need remains for a practical model for determining actualand potential energy consumption for the heating and cooling of abuilding.

A further need remains for an approach to quantifying improvements inenergy consumption and cost savings resulting from building shellupgrades.

SUMMARY

Fuel consumption for building heating and cooling can be calculatedthrough two practical approaches that characterize a building's thermalefficiency through empirically-measured values and readily-obtainableenergy consumption data, such as available in utility bills, therebyavoiding intrusive and time-consuming analysis with specialized testingequipment. While the discussion is herein centered on building heatingrequirements, the same principles can be applied to an analysis ofbuilding cooling requirements. The first approach can be used tocalculate annual or periodic fuel requirements. The approach requiresevaluating typical monthly utility billing data and approximations ofheating (or cooling) losses and gains.

The second approach can be used to calculate hourly (or interval) fuelrequirements. The approach includes empirically deriving threebuilding-specific parameters: thermal mass, thermal conductivity, andeffective window area. HVAC system power rating and conversion anddelivery efficiency are also parametrized. The parameters are estimatedusing short duration tests that last at most several days. Theparameters and estimated HVAC system efficiency are used to simulate atime series of indoor building temperature. In addition, the secondhourly (or interval) approach can be used to verify or explain theresults from the first annual (or periodic) approach. For instance, timeseries results can be calculated using the second approach over the spanof an entire year and compared to results determined through the firstapproach. Other uses of the two approaches and forms of comparison arepossible.

The empirically derived building-specific parameters can beformulaically applied to determine seasonal fuel consumption and indoortemperatures pertaining to the heating season, the summer or year round.To generalize the approach as set forth for winter, the term used torepresent furnace rating is replaced with HVAC system rating. As totalheat change is based on the building's thermal mass, heat change isrelabeled as thermal mass gain (or loss). This change creates a heatbalance equation that is composed of heat gain (loss) from six sources,three of which contribute to heat gain only. No modifications arerequired for applying the empirical tests to summer since an attic'sthermal conductivity cancels out and the attic's effective window areais directly combined with the existing effective window area. Moreover,since these tests are empirically based, the tests already account forthe additional heat gain associated with the elevated attic temperatureand other surface temperatures.

Furthermore, the net savings in fuel, cost and carbon emissions(environmental) provided by an investment in an electric energyefficiency can be determined. The net effect that a proposed electricenergy efficiency investment has in reductions in the fuel consumed fora building's heating and cooling can be weighed by taking into accountthe efficiencies of electricity generation as supplied to a building andof the building's cooling and heating systems, fuel costs, and carbonemissions.

One embodiment provides a system and method estimating indoortemperature time series data of a building with the aid of a digitalcomputer. Thermal conductivity, thermal mass, effective window area, andHVAC system efficiency of a building are found with involvement of acomputer through empirical testing of the building over a monitored timeframe. A difference between indoor and outdoor temperatures of thebuilding during the monitored time frame are recorded with involvementof the computer. A time period for each interval in a time series isdefined. An average occupancy, internal electricity consumption, solarresource, HVAC system rating, and HVAC system status as applicable tothe building are retrieved over the monitored time frame. The timeseries, including temperature data based on the building's indoortemperature, thermal mass, thermal conductivity, temperature difference,occupancy, internal electricity consumption, effective window area,solar resource, and the rating, efficiency and status of the HVACsystem, is built.

A further embodiment provides a system and method for estimatingperiodic fuel consumption for cooling of a building with the aid of adigital computer. Thermal conductivity, thermal mass, effective windowarea, and HVAC system efficiency of a building are found withinvolvement of a computer through empirical testing of the building overa monitored time frame. A difference between indoor and outdoortemperatures of the building is recorded with involvement of thecomputer during the monitored time frame. A time span over which toestimate is defined with the computer. An average occupancy, averageinternal electricity consumption, average solar resource, HVAC systemrating, and HVAC system status as applicable to the building over themonitored time frame is retrieved with the computer. The periodic fuelconsumption for cooling over the modeled time span is estimated based onthe building's thermal conductivity, temperature difference, number oftime periods comprised in the modeled time span, average occupancy,average internal electricity consumption, effective window area, andaverage solar resource internal electricity consumption internalelectricity consumption.

A further embodiment provides a system and method for estimatingseasonal net fuel savings with the aid of a digital computer.Efficiencies of electricity generation as supplied to a building and ofthe building's cooling and heating systems are obtained with a computer.A cooling season duration and a heating season duration that togethercomprise seasonal changes affecting the building are defined with thecomputer. A net fuel savings afforded by an electric energy efficiencyis evaluated as a function of a reduction in electricity consumptionafforded by the electric energy efficiency times an inverse of theelectricity generation efficiency less an inverse of the heating systemefficiency based on the heating season duration plus an inverse of thecooling system efficiency based on the cooling season duration.

A further embodiment provides a system and method for estimatingseasonal net cost savings with the aid of a digital computer.Efficiencies of electricity generation as supplied to a building and ofthe building's cooling and heating systems are obtained with a computer.Prices of electricity and natural gas are obtained with the computer. Acooling season duration and a heating season duration that togethercomprise seasonal changes affecting the building are defined with thecomputer. A net cost savings afforded by an electric energy efficiencyis evaluated as a function of a reduction in electricity consumptionafforded by the electric energy efficiency times the electricity priceplus an inverse of the cooling system efficiency based on the coolingseason duration less the natural gas price over the heating systemefficiency based on the heating season duration.

A further embodiment provides a system and method for estimatingseasonal net carbon emissions savings with the aid of a digitalcomputer. Efficiencies of electricity generation as supplied to abuilding and of the building's cooling and heating systems are obtainedwith a computer. Carbon emissions of electricity and natural gasconsumption are obtained with the computer. A cooling season durationand a heating season duration that together comprise seasonal changesaffecting the building are defined with the computer. A net carbonemissions savings afforded by an electric energy efficiency is evaluatedas a function of a reduction in electricity consumption afforded by theelectric energy efficiency times the electricity consumption carbonemissions plus an inverse of the cooling system efficiency based on thecooling season duration less the natural gas consumption carbonemissions over the heating system efficiency based on the heating seasonduration.

The foregoing approaches, annual (or periodic) and hourly (or interval)improve upon and compliment the standard energy audit-style methodologyof estimating heating (and cooling) fuel consumption in several ways.First, per the first approach, the equation to calculate annual fuelconsumption and its derivatives is simplified over thefully-parameterized form of the equation used in energy audit analysis,yet without loss of accuracy. Second, both approaches require parametersthat can be obtained empirically, rather than from a detailed energyaudit that requires specialized testing equipment and prescribed testconditions. Third, per the second approach, a time series of indoortemperature and fuel consumption data can be accurately generated. Theresulting fuel consumption data can then be used by economic analysistools using prices that are allowed to vary over time to quantifyeconomic impact.

Moreover, the economic value of heating (and cooling) energy savingsassociated with any building shell improvement in any building has beenshown to be independent of building type, age, occupancy, efficiencylevel, usage type, amount of internal electric gains, or amount solargains, provided that fuel has been consumed at some point for auxiliaryheating. The only information required to calculate savings includes thenumber of hours that define the winter season; average indoortemperature; average outdoor temperature; the building's HVAC systemefficiency (or coefficient of performance for heat pump systems); thearea of the existing portion of the building to be upgraded; the R-valueof the new and existing materials; and the average price of energy, thatis, heating fuel.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 2 is a graph showing, by way of example, balance point thermalconductivity.

FIG. 3 is a flow diagram showing a computer-implemented method formodeling periodic building heating energy consumption in accordance withone embodiment.

FIG. 4 is a flow diagram showing a routine for determining heating gainsfor use in the method of FIG. 3.

FIG. 5 is a flow diagram showing a routine for balancing energy for usein the method of FIG. 3.

FIG. 6 is a process flow diagram showing, by way of example, consumerheating energy consumption-related decision points.

FIG. 7 is a table showing, by way of example, data used to calculatethermal conductivity.

FIG. 8 is a table showing, by way of example, thermal conductivityresults for each season using the data in the table of FIG. 7 as inputsinto Equations (28) through (31).

FIG. 9 is a graph showing, by way of example, a plot of the thermalconductivity results in the table of FIG. 8.

FIG. 10 is a graph showing, by way of example, an auxiliary heatingenergy analysis and energy consumption investment options.

FIG. 11 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 12 is a flow diagram showing a computer-implemented method formodeling interval building heating energy consumption in accordance witha further embodiment.

FIG. 13 is a table showing the characteristics of empirical tests usedto solve for the four unknown parameters in Equation (50).

FIG. 14 is a flow diagram showing a routine for empirically determiningbuilding- and equipment-specific parameters using short duration testsfor use in the method of FIG. 12.

FIG. 15 is a graph showing, by way of example, measured and modeledhalf-hour indoor temperature from June 15 to Jul. 30, 2015.

FIG. 16 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach.

FIG. 17 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the allowable indoor temperaturelimited to 2° F. above desired temperature of 68° F.

FIG. 18 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the size of effective window areatripled from 2.5 m² to 7.5 m².

FIG. 19 is a table showing, by way of example, test data.

FIG. 20 is a table showing, by way of example, the statistics performedon the data in the table of FIG. 19 required to calculate the three testparameters.

FIG. 21 is a graph showing, by way of example, hourly indoor (measuredand simulated) and outdoor (measured) temperatures.

FIG. 22 is a graph showing, by way of example, simulated versus measuredhourly temperature delta (indoor minus outdoor).

FIG. 23 is a block diagram showing a computer-implemented system formodeling building heating energy consumption in accordance with oneembodiment.

DETAILED DESCRIPTION

Conventional Energy Audit-Style Approach

Conventionally, estimating periodic HVAC energy consumption andtherefore fuel costs includes analytically determining a building'sthermal conductivity (UA^(Total)) based on results obtained through anon-site energy audit. For instance, J. Randolf and G. Masters, Energyfor Sustainability: Technology, Planning, Policy, pp. 247, 248, 279(2008), present a typical approach to modeling heating energyconsumption for a building, as summarized by Equations 6.23, 6.27, and7.5. The combination of these equations states that annual heating fuelconsumption Q^(Fuel) equals the product of UA^(Total), 24 hours per day,and the number of heating degree days (HDD) associated with a particularbalance point temperature T^(Balance Point), as adjusted for the solarsavings fraction (SSF) divided by HVAC system efficiency (η^(HVAC)):

$\begin{matrix}{Q^{Fuel} = {\left( {UA}^{Total} \right)\left( {24*{HDD}^{T^{{Balance}\mspace{14mu}{Point}}}} \right)\left( {1 - {SSF}} \right)\left( \frac{1}{\eta^{HVAC}} \right)}} & (1)\end{matrix}$such that:

$\begin{matrix}{T^{{Balance}\mspace{14mu}{Point}} = {T^{{Set}\mspace{14mu}{Point}} - \frac{{Internal}\mspace{14mu}{Gains}}{{UA}^{Total}}}} & (2)\end{matrix}$andη^(HVAC)=η_(Furnace)η_(Distribution)  (3)where T^(SetPoint) represents the temperature setting of the thermostat,Internal Gains represents the heating gains experienced within thebuilding as a function of heat generated by internal sources andauxiliary heating, as further discussed infra, η^(Furnace) representsthe efficiency of the furnace or heat source proper, andη^(Distribution) represents the efficiency of the duct work and heatdistribution system. For clarity, HDD^(T) ^(Balance Point) will beabbreviated to HDD^(Balance Point Temp).

A cursory inspection of Equation (1) implies that annual fuelconsumption is linearly related to a building's thermal conductivity.This implication further suggests that calculating fuel savingsassociated with building envelope or shell improvements isstraightforward. In practice, however, such calculations are notstraightforward because Equation (1) was formulated with the goal ofdetermining the fuel required to satisfy heating energy needs. As such,there are several additional factors that the equation must take intoconsideration.

First, Equation (1) needs to reflect the fuel that is required only whenindoor temperature exceeds outdoor temperature. This need led to theheating degree day (HDD) approach (or could be applied on a shorter timeinterval basis of less than one day) of calculating the differencebetween the average daily (or hourly) indoor and outdoor temperaturesand retaining only the positive values. This approach complicatesEquation (1) because the results of a non-linear term must be summed,that is, the maximum of the difference between average indoor andoutdoor temperatures and zero. Non-linear equations complicateintegration, that is, the continuous version of summation.

Second, Equation (1) includes the term Balance Point temperature(T^(Balance Point)). The goal of including the term T^(Balance Point)was to recognize that the internal heating gains of the buildingeffectively lowered the number of degrees of temperature that auxiliaryheating needed to supply relative to the temperature setting of thethermostat T^(Set Point). A balance point temperature T^(Balance Point)of 65° F. was initially selected under the assumption that 65° F.approximately accounted for the internal gains. As buildings became moreefficient, however, an adjustment to the balance point temperatureT^(Balance Point) was needed based on the building's thermalconductivity (UA^(Total)) and internal gains. This further complicatedEquation (1) because the equation became indirectly dependent on (andinversely related to) UA^(Total) through T^(Balance Point).

Third, Equation (1) addresses fuel consumption by auxiliary heatingsources. As a result, Equation (1) must be adjusted to account for solargains. This adjustment was accomplished using the Solar Savings Fraction(SSF). The SSF is based on the Load Collector Ratio (see Eq. 7.4 inRandolf and Masters, p. 278, cited supra, for information about theLCR). The LCR, however, is also a function of UA^(Total). As a result,the SSF is a function of UA^(Total) in a complicated, non-closed formsolution manner. Thus, the SSF further complicates calculating the fuelsavings associated with building shell improvements because the SSF isindirectly dependent on UA^(Total).

As a result, these direct and indirect dependencies significantlycomplicate calculating a change in annual fuel consumption based on achange in thermal conductivity. The difficulty is made evident by takingthe derivative of Equation (1) with respect to a change in thermalconductivity. The chain and product rules from calculus need to beemployed since HDD^(Balance Point Temp) and SSF are indirectly dependenton UA^(Total):

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = {\left\{ {{\left( {UA}^{Total} \right)\left\lbrack {{\left( {HDD}^{{Balance}\mspace{14mu}{Point}\mspace{14mu}{Temp}} \right)\left( {{- \frac{dSSF}{dLCR}}\frac{dLCR}{{dUA}^{Total}}} \right)} + {\left( {\frac{{dHDD}^{{Balance}\mspace{14mu}{Point}\mspace{14mu}{Temp}}}{{dT}^{{Balance}\mspace{14mu}{Point}}}\frac{{dT}^{{Balance}\mspace{14mu}{Point}}}{{dUA}^{Total}}} \right)\left( {1 - {SSF}} \right)}} \right\rbrack} + {\left( {HDD}^{{Balance}\mspace{14mu}{Point}\mspace{14mu}{Temp}} \right)\left( {1 - {SSF}} \right)}} \right\}\left( \frac{24}{\eta^{HVAC}} \right)}} & (4)\end{matrix}$The result is Equation (4), which is an equation that is difficult tosolve due to the number and variety of unknown inputs that are required.

To add even further complexity to the problem of solving Equation (4),conventionally, UA^(Total) is determined analytically by performing adetailed energy audit of a building. An energy audit involves measuringphysical dimensions of walls, windows, doors, and other building parts;approximating R-values for thermal resistance; estimating infiltrationusing a blower door test; and detecting air leakage. A numerical modelis then run to perform the calculations necessary to estimate thermalconductivity. Such an energy audit can be costly, time consuming, andinvasive for building owners and occupants. Moreover, as a calculatedresult, the value estimated for UA^(Total) carries the potential forinaccuracies, as the model is strongly influenced by physicalmismeasurements or omissions, data assumptions, and so forth.

Empirically-Based Approaches to Modeling Heating Fuel Consumption

Building heating (and cooling) fuel consumption can be calculatedthrough two approaches, annual (or periodic) and hourly (or interval) tothermally characterize a building without intrusive and time-consumingtests. The first approach, as further described infra beginning withreference to FIG. 1, requires typical monthly utility billing data andapproximations of heating (or cooling) losses and gains. The secondapproach, as further described infra beginning with reference to FIG.11, involves empirically deriving three building-specific parameters,thermal mass, thermal conductivity, and effective window area, plus HVACsystem efficiency using short duration tests that last at most severaldays. The parameters are then used to simulate a time series of indoorbuilding temperature and of fuel consumption. While the discussionherein is centered on building heating requirements, the same principlescan be applied to an analysis of building cooling requirements. Inaddition, conversion factors for occupant heating gains (250 Btu of heatper person per hour), heating gains from internal electricityconsumption (3,412 Btu per kWh), solar resource heating gains (3,412 Btuper kWh), and fuel pricing (

$\frac{{Price}^{NG}}{10^{5}}$it in units or $ per therm and

$\frac{{Price}^{Electrity}}{3\text{,}412}$if in units of $ per kWh) are used by way of example; other conversionfactors or expressions are possible.First Approach: Annual (or Periodic) Fuel Consumption

Fundamentally, thermal conductivity is the property of a material, here,a structure, to conduct heat. FIG. 1 is a functional block diagram 10showing heating losses and gains relative to a structure 11.Inefficiencies in the shell 12 (or envelope) of a structure 11 canresult in losses in interior heating 14, whereas gains 13 in heatinggenerally originate either from sources within (or internal to) thestructure 11, including heating gains from occupants 15, gains fromoperation of electric devices 16, and solar gains 17, or from auxiliaryheating sources 18 that are specifically intended to provide heat to thestructure's interior.

In this first approach, the concepts of balance point temperatures andsolar savings fractions, per Equation (1), are eliminated. Instead,balance point temperatures and solar savings fractions are replaced withthe single concept of balance point thermal conductivity. Thissubstitution is made by separately allocating the total thermalconductivity of a building (UA^(Total)) to thermal conductivity forinternal heating gains (UA^(BalancePoint)), including occupancy, heatproduced by operation of certain electric devices, and solar gains, andthermal conductivity for auxiliary heating (UA^(Auxilary Heating)). Theend result is Equation (34), further discussed in detail infra, whicheliminates the indirect and non-linear parameter relationships inEquation (1) to UA^(Total).

The conceptual relationships embodied in Equation (34) can be describedwith the assistance of a diagram. FIG. 2 is a graph 20 showing, by wayof example, balance point thermal conductivity UA^(Balance Point), thatis, the thermal conductivity for internal heating gains. The x-axis 21represents total thermal conductivity, UA^(Total), of a building (inunits of Btu/hr-° F.). The y-axis 22 represents total heating energyconsumed to heat the building. Total thermal conductivity 21 (along thex-axis) is divided into “balance point” thermal conductivity(UA^(Balance Point)) 23 and “heating system” (or auxiliary heating)thermal conductivity (UA^(Auxiliary Heating)) 24. “Balance point”thermal conductivity 23 characterizes heating losses, which can occur,for example, due to the escape of heat through the building envelope tothe outside and by the infiltration of cold air through the buildingenvelope into the building's interior that are compensated for byinternal gains. “Heating system” thermal conductivity 24 characterizesheating gains, which reflects the heating delivered to the building'sinterior above the balance point temperature T^(Balance Point),generally as determined by the setting of the auxiliary heating source'sthermostat or other control point.

In this approach, total heating energy 22 (along the y-axis) is dividedinto gains from internal heating 25 and gains from auxiliary heatingenergy 25. Internal heating gains are broken down into heating gainsfrom occupants 27, gains from operation of electric devices 28 in thebuilding, and solar gains 29. Sources of auxiliary heating energyinclude, for instance, natural gas furnace 30 (here, with a 56%efficiency), electric resistance heating 31 (here, with a 100%efficiency), and electric heat pump 32 (here, with a 250% efficiency).Other sources of heating losses and gains are possible.

The first approach provides an estimate of fuel consumption over a yearor other period of inquiry based on the separation of thermalconductivity into internal heating gains and auxiliary heating. FIG. 3is a flow diagram showing a computer-implemented method 40 for modelingperiodic building heating energy consumption in accordance with oneembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 23, as a series of process or method modules or steps.

In the first part of the approach (steps 41-43), heating losses andheating gains are separately analyzed. In the second part of theapproach (steps 44-46), the portion of the heating gains that need to beprovided by fuel, that is, through the consumption of energy forgenerating heating using auxiliary heating 18 (shown in FIG. 1), isdetermined to yield a value for annual (or periodic) fuel consumption.Each of the steps will now be described in detail.

Specify Time Period

Heating requirements are concentrated during the winter months, so as aninitial step, the time period of inquiry is specified (step 41). Theheating degree day approach (HDD) in Equation (1) requires examining allof the days of the year and including only those days where outdoortemperatures are less than a certain balance point temperature. However,this approach specifies the time period of inquiry as the winter seasonand considers all of the days (or all of the hours, or other time units)during the winter season. Other periods of inquiry are also possible,such as a five- or ten-year time frame, as well as shorter time periods,such as one- or two-month intervals.

Separate Heating Losses from Heating Gains

Heating losses are considered separately from heating gains (step 42).The rationale for drawing this distinction will now be discussed.

Heating Losses

For the sake of discussion herein, those regions located mainly in thelower latitudes, where outdoor temperatures remain fairly moderate yearround, will be ignored and focus placed instead on those regions thatexperience seasonal shifts of weather and climate. Under thisassumption, a heating degree day (HDD) approach specifies that outdoortemperature must be less than indoor temperature. No such limitation isapplied in this present approach. Heating losses are negative if outdoortemperature exceeds indoor temperature, which indicates that thebuilding will gain heat during these times. Since the time period hasbeen limited to only the winter season, there will likely to be alimited number of days when that situation could occur and, in thoselimited times, the building will benefit by positive heating gain. (Notethat an adjustment would be required if the building took advantage ofthe benefit of higher outdoor temperatures by circulating outdoor airinside when this condition occurs. This adjustment could be made bytreating the condition as an additional source of heating gain.)

As a result, fuel consumption for heating losses Q^(Losses) over thewinter season equals the product of the building's total thermalconductivity UA^(Total) and the difference between the indoor T^(Indoor)and outdoor temperature T^(Outdoor), summed over all of the hours of thewinter season:

$\begin{matrix}{Q^{Losses} = {\sum\limits_{t^{Start}}^{t^{End}}{\left( {UA}^{Total} \right)\left( {T_{t}^{Indoor} - T_{t}^{Outdoor}} \right)}}} & (5)\end{matrix}$where Start and End respectively represent the first and last hours ofthe winter (heating) season.

Equation (5) can be simplified by solving the summation. Thus, totalheating losses Q^(Losses) equal the product of thermal conductivityUA^(Total) and the difference between average indoor temperature T^(Indoor) and average outdoor temperature T ^(Outdoor) over the winterseason and the number of hours H in the season over which the average iscalculated:Q ^(Losses)=(UA ^(Total))( T ^(Indoor) −T ^(Outdoor))(H)  (6)

Heating Gains

Heating gains are calculated for two broad categories (step 43) based onthe source of heating, internal heating gains Q^(Gains-Internal) andauxiliary heating gains Q^(Gains-Auxiliary Heating), as furtherdescribed infra with reference to FIG. 4. Internal heating gains can besubdivided into heating gained from occupants Q^(Gains-Occupants),heating gained from the operation of electric devices Q^(Gains-Electric)and heating gained from solar heating Q^(Gains-Solar). Other sources ofinternal heating gains are possible. The total amount of heating gainedQ^(Gains) from these two categories of heating sources equals:Q ^(Gains) =Q ^(Gains-Internal) +Q ^(Gains-Auxiliary Heating)  (7)whereQ ^(Gains-Internal) =Q ^(Gains-Occupants) +Q ^(Gains-Electric) +Q^(Gains-Solar)  (8)Calculate Heating Gains

Equation (8) states that internal heating gains Q^(Gains-Internal)include heating gains from Occupant, Electric, and Solar heatingsources. FIG. 4 is a flow diagram showing a routine 50 for determiningheating gains for use in the method 40 of FIG. 3 Each of these heatinggain sources will now be discussed.

Occupant Heating Gains

People occupying a building generate heat. Occupant heating gainsQ^(Gains-Occupants) (step 51) equal the product of the heat produced perperson, the average number of people in a building over the time period,and the number of hours (H) (or other time units) in that time period.Let P represent the average number of people. For instance, using aconversion factor of 250 Btu of heat per person per hour, heating gainsfrom the occupants Q^(Gains-Occupants) equal:Q ^(Gains-Occupants)=250( P )(H)  (9)Other conversion factors or expressions are possible.

Electric Heating Gains

The operation of electric devices that deliver all heat that isgenerated into the interior of the building, for instance, lights,refrigerators, and the like, contribute to internal heating gain.Electric heating gains Q^(Gains-Electric) (step 52) equal the amount ofelectricity used in the building that is converted to heat over the timeperiod.

Care needs to be taken to ensure that the measured electricityconsumption corresponds to the indoor usage. Two adjustments may berequired. First, many electric utilities measure net electricityconsumption. The energy produced by any photovoltaic (PV) system needsto be added back to net energy consumption (Net) to result in grossconsumption if the building has a net-metered PV system. This amount canbe estimated using time- and location-correlated solar resource data, aswell as specific information about the orientation and othercharacteristics of the photovoltaic system, such as can be provided bythe Solar Anywhere SystemCheck service (http://www.SolarAnywhere.com), aWeb-based service operated by Clean Power Research, L.L.C., Napa,Calif., with the approach described, for instance, in commonly-assignedU.S. patent application, entitled “Computer-Implemented System andMethod for Estimating Gross Energy Load of a Building,” Ser. No.14/531,940, filed Nov. 3, 2014, pending, the disclosure of which isincorporated by reference, or measured directly.

Second, some uses of electricity may not contribute heat to the interiorof the building and need be factored out as external electric heatinggains (External). These uses include electricity used for electricvehicle charging, electric dryers (assuming that most of the hot exhaustair is vented outside of the building, as typically required by buildingcode), outdoor pool pumps, and electric water heating using eitherdirect heating or heat pump technologies (assuming that most of the hotwater goes down the drain and outside the building—a large body ofstanding hot water, such as a bathtub filled with hot water, can beconsidered transient and not likely to appreciably increase thetemperature indoors over the long run).

For instance, using a conversion factor from kWh to Btu of 3,412 Btu perkWh (since Q^(Gains-Electric) is in units of Btu), internal electricgains Q^(Gains-Electric) equal:

$\begin{matrix}{Q^{{Gains} - {Electric}} = {\left( \overset{\_}{{Net} + {PV} - {External}} \right)(H)\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{kWh} \right)}} & (10)\end{matrix}$where Net represents net energy consumption, PV represents any energyproduced by a PV system, External represents heating gains attributableto electric sources that do not contribute heat to the interior of abuilding. Other conversion factors or expressions are possible. Theaverage delivered electricity Net+PV−External equals the total over thetime period divided by the number of hours (H) in that time period.

$\begin{matrix}{\overset{\_}{{Net} + {PV} - {External}} = \frac{{Net} + {PV} - {External}}{H}} & (11)\end{matrix}$

Solar Heating Gains

Solar energy that enters through windows, doors, and other openings in abuilding as sunlight will heat the interior. Solar heating gainsQ^(Gains-Solar) (step 53) equal the amount of heat delivered to abuilding from the sun. In the northern hemisphere, Q^(Gains-Solar) canbe estimated based on the south-facing window area (m²) times the solarheating gain coefficient (SHGC) times a shading factor; together, theseterms are represented by the effective window area (W). Solar heatinggains Q^(Gains-Solar) equal the product of W, the average directvertical irradiance (DVI) available on a south-facing surface (Solar, asrepresented by DVI in kW/m²), and the number of hours (H) in the timeperiod. For instance, using a conversion factor from kWh to Btu of 3,412Btu per kWh (since Q^(Gains-Solar) is in units of Btu while averagesolar is in kW/m²⁾, solar heating gains Q^(Gai ns-Solar) equal:

$\begin{matrix}{Q^{{Gains} - {Solar}} = {\left( \overset{\_}{Solar} \right)(W)(H)\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{kWh} \right)}} & (12)\end{matrix}$Other conversion factors or expressions are possible.

Note that for reference purposes, the SHGC for one particular highquality window designed for solar gains, the Andersen High-PerformanceLow-E4 PassiveSun Glass window product, manufactured by AndersenCorporation, Bayport, Minn., is 0.54; many windows have SHGCs that arebetween 0.20 to 0.25.

Auxiliary Heating Gains

The internal sources of heating gain share the common characteristic ofnot being operated for the sole purpose of heating a building, yetnevertheless making some measureable contribution to the heat to theinterior of a building. The fourth type of heating gain, auxiliaryheating gains Q^(Gains-Auxiliary Heating), consumes fuel specifically toprovide heat to the building's interior and, as a result, must includeconversion efficiency. The gains from auxiliary heating gainsQ^(Gains-Auxiliary H eating)(step 53) equal the product of the averagehourly fuel consumed Q ^(Fuel) times the hours (H) in the period timesHVAC system efficiency η^(HVAC).Q ^(Gains-Auxiliary Heating)=( Q ^(Fuel))(H)(η^(HVAC))  (13)

Equation (13) can be stated in a more general form that can be appliedto both heating and cooling seasons by adding a binary multiplier,HeatOrCool. The binary multiplier HeatOrCool equals 1 when the heatingsystem is in operation and equals −1 when the cooling system is inoperation. This more general form will be used in a subsequent section.Q ^(Gains(Losses)-HVAC)=(HeatOrCool)( Q ^(Fuel))(H)(η^(HVAC))  (14)

Divide Thermal Conductivity into Parts

Consider the situation when the heating system is in operation. TheHeatingOrCooling term in Equation (14) equals 1 in the heating season.As illustrated in FIG. 3, a building's thermal conductivity UA^(Total),rather than being treated as a single value, can be conceptually dividedinto two parts (step 44), with a portion of UA^(Total) allocated to“balance point thermal conductivity” (UA^(Balance Point)) and a portionto “auxiliary heating thermal conductivity” (UA^(Auxilary Heating)),such as pictorially described supra with reference to FIG. 2.UA^(Balance Point) corresponds to the heating losses that a building cansustain using only internal heating gains Q^(Gains-Internal). This valueis related to the concept that a building can sustain a specifiedbalance point temperature in light of internal gains. However, insteadof having a balance point temperature, some portion of the buildingUA^(Balance Point) is considered to be thermally sustainable givenheating gains from internal heating sources (Q^(Gains-Internal)). As therest of the heating losses must be made up by auxiliary heating gains,the remaining portion of the building UA^(Auxilary Heating) isconsidered to be thermally sustainable given heating gains fromauxiliary heating sources (Q^(Gains-Auxiblay Heating)). The amount ofauxiliary heating gained is determined by the setting of the auxiliaryheating source's thermostat or other control point. Thus, UA^(Total) canbe expressed as:UA ^(Total) =UA ^(Balance Point) +UA ^(Auxiliary Heating)  (15)whereUA ^(Balance Point) =UA ^(Occupants) +UA ^(Electric) +UA ^(Solar)  (16)such that UA^(Occupants), UA^(Electric), and UA^(Solar) respectivelyrepresent the thermal conductivity of internal heating sources,specifically, occupants, electric and solar.

In Equation (15), total thermal conductivity UA^(Total) is fixed at acertain value for a building and is independent of weather conditions;UA^(Total) depends upon the building's efficiency. The component partsof Equation (15), balance point thermal conductivity UA^(Balance Point)and auxiliary heating thermal conductivity UA^(Auxiliary Heating),however, are allowed to vary with weather conditions. For example, whenthe weather is warm, there may be no auxiliary heating in use and all ofthe thermal conductivity will be allocated to the balance point thermalconductivity UA^(Balance Point) component.

Fuel consumption for heating losses Q^(Losses) can be determined bysubstituting Equation (15) into Equation (6):Q ^(Losses)=(UA ^(Balance Point) +UA ^(Auxiliary Heating))( T ^(Indoor)−T ^(Outdoor))(H)  (17)

Balance Energy

Heating gains must equal heating losses for the system to balance (step45), as further described infra with reference to FIG. 5. Heating energybalance is represented by setting Equation (7) equal to Equation (17):Q ^(Gains-Internal) +Q ^(Gains-Auxiliary Heating)=(UA ^(Balance Point)+UA ^(Auxiliary Heating))( T ^(Indoor) −T ^(Outdoor))(H)  (18)The result can then be divided by (T ^(Indoor)−T ^(Outdoor))(H),assuming that this term is non-zero:

$\begin{matrix}{{{UA}^{{Balance}\mspace{11mu}{Point}} + {UA}^{{Auxiliary}\mspace{11mu}{Heating}}} = \frac{Q^{{Gains} - {Internal}} + Q^{{Gains} - {{Auxiliary}\mspace{11mu}{Heating}}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (19)\end{matrix}$

Equation (19) expresses energy balance as a combination of bothUA^(Balance Point) and UA^(Auxilary Heating). FIG. 5 is a flow diagramshowing a routine 60 for balancing energy for use in the method 40 ofFIG. 3. Equation (19) can be further constrained by requiring that thecorresponding terms on each side of the equation match, which willdivide Equation (19) into a set of two equations:

$\begin{matrix}{{UA}^{{Balance}\mspace{11mu}{Point}} = \frac{Q^{{Gains} - {Internal}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (20) \\{{UA}^{{Auxiliary}\mspace{11mu}{Heating}} = \frac{Q^{{Gains} - {{Auxiliary}\mspace{11mu}{Heating}}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (21)\end{matrix}$

The UA^(Balance Point) should always be a positive value. Equation (20)accomplishes this goal in the heating season. An additional term,HeatOrCool is required for the cooling season that equals 1 in theheating season and −1 in the cooling season.

$\begin{matrix}{{UA}^{{Balance}\mspace{11mu}{Point}} = \frac{({HeatOrCool})\left( Q^{{Gains} - {Internal}} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (22)\end{matrix}$

HeatOrCool and its inverse are the same. Thus, internal gains equals:(23)Q ^(Gains-Internal)=(HeatOrCool)(UA ^(Balance Point))( T ^(Indoor) −T^(Outdoor))(H)  (23)

Components of UA^(Balance Point)

For clarity, UA^(Balance Point) can be divided into three componentvalues (step 61) by substituting Equation (8) into Equation (20):

$\begin{matrix}{{UA}^{{Balance}\mspace{11mu}{Point}} = \frac{Q^{{Gains} - {Occupants}} + Q^{{Gains} - {Electric}} + Q^{{Gains} - {Solar}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (24)\end{matrix}$

Since UA^(Balance Point) equals the sum of three component values (asspecified in Equation (16)), Equation (24) can be mathematically limitedby dividing Equation (24) into three equations:

$\begin{matrix}{{UA}^{Occupants} = \frac{Q^{{Gains} - {Occupants}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (25) \\{{UA}^{Electric} = \frac{Q^{{Gains} - {Electric}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (26) \\{{UA}^{Solar} = \frac{Q^{{Gains} - {Solar}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (27)\end{matrix}$

Solutions for Components of UA^(Balance Point) andUA^(Auxiliary Heating)

The preceding equations can be combined to present a set of results withsolutions provided for the four thermal conductivity components asfollows. First, the portion of the balance point thermal conductivityassociated with occupants UA^(Occupants) (step 62) is calculated bysubstituting Equation (9) into Equation (25). Next, the portion of thebalance point thermal conductivity UA^(Electric) associated withinternal electricity consumption (step 63) is calculated by substitutingEquation (10) into Equation (26). Internal electricity consumption isthe amount of electricity consumed internally in the building andexcludes electricity consumed for HVAC operation, pool pump operation,electric water heating, electric vehicle charging, and so on, sincethese sources of electricity consumption result in heat or work beingused external to the inside of the building. The portion of the balancepoint thermal conductivity UA^(solar) associated with solar gains (step64) is then calculated by substituting Equation (12) into Equation (27).Finally, thermal conductivity UA^(Auxiliary Heating) associated withauxiliary heating (step 64) is calculated by substituting Equation (13)into Equation (21).

$\begin{matrix}{{UA}^{Occupants} = \frac{250\left( \overset{\_}{P} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (28) \\{{UA}^{Electric} = {\frac{\left( \overset{\_}{{Net} + {PV} + {External}} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{kWh} \right)}} & (29) \\{{UA}^{Solar} = {\frac{\left( \overset{\_}{Solar} \right)(W)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{kWh} \right)}} & (30) \\{{UA}^{{Auxiliary}\mspace{11mu}{Heating}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (31)\end{matrix}$

Determine Fuel Consumption

Referring back to FIG. 3, Equation (31) can used to derive a solution toannual (or periodic) heating fuel consumption. First, Equation (15) issolved for UA^(Auxilary Heating):UA ^(Auxiliary Heating) =UA ^(Total) −UA ^(Balance Point)  (32)Equation (32) is then substituted into Equation (31):

$\begin{matrix}{{{UA}^{Total} - {UA}^{{Balance}\mspace{11mu}{Point}}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (33)\end{matrix}$Finally, solving Equation (33) for fuel and multiplying by the number ofhours (H) in (or duration of) the time period yields:

${Q^{Fuel} = {\frac{\left( {{UA}^{Total} - {UA}^{{Balance}\mspace{11mu}{Point}}} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}\mspace{70mu}(34)}}\mspace{14mu}$Equation (34) is valid during the heating season and applies whereUA^(Total)≥UA^(Balance Point). Otherwise, fuel consumption is 0.

Using Equation (34), annual (or periodic) heating fuel consumptionQ^(Fuel) can be determined (step 46). The building's thermalconductivity UA^(Total), if already available through, for instance, theresults of an energy audit, is obtained. Otherwise, UA^(Total) can bedetermined by solving Equations (28) through (31) using historical fuelconsumption data, such as shown, by way of example, in the table of FIG.7, or by solving Equation (52), as further described infra. UA^(Total)can also be empirically determined with the approach described, forinstance, in commonly-assigned U.S. patent application, entitled “Systemand Method for Empirically Estimating Overall Thermal Performance of aBuilding,” Ser. No. 14/294,087, filed Jun. 2, 2014, pending, thedisclosure of which is incorporated by reference. Other ways todetermine UA^(Total) are possible. UA^(Balance Point) can be determinedby solving Equation (24). The remaining values, average indoortemperature T ^(Indoor) and average outdoor temperature T ^(Outdoor) andHVAC system efficiency η^(HVAC), can respectively be obtained fromhistorical weather data and manufacturer specifications.

Practical Considerations

Equation (34) is empowering. Annual heating fuel consumption Q^(Fuel)can be readily determined without encountering the complications ofEquation (1), which is an equation that is difficult to solve due to thenumber and variety of unknown inputs that are required. The implicationsof Equation (34) in consumer decision-making, a general discussion, andsample applications of Equation (34) will now be covered.

Change in Fuel Requirements Associated with Decisions Available toConsumers

Consumers have four decisions available to them that affects theirenergy consumption for heating. FIG. 6 is a process flow diagramshowing, by way of example, consumer heating energy consumption-relateddecision points. These decisions 71 include:

-   -   1. Change the thermal conductivity UA^(Total) by upgrading the        building shell to be more thermally efficient (process 72).    -   2. Reduce or change the average indoor temperature by reducing        the thermostat manually, programmatically, or through a        “learning” thermostat (process 73).    -   3. Upgrade the HVAC system to increase efficiency (process 74).    -   4. Increase the solar gain by increasing the effective window        area (process 75).        Other decisions are possible. Here, these four specific options        can be evaluated supra by simply taking the derivative of        Equation (34) with respect to a variable of interest. The result        for each case is valid where UA^(Total)≥UA^(Balance Point).        Otherwise, fuel consumption is 0.

Changes associated with other internal gains, such as increasingoccupancy, increasing internal electric gains, or increasing solarheating gains, could be calculated using a similar approach.

Change in Thermal Conductivity

A change in thermal conductivity UA^(Total) can affect a change in fuelrequirements. The derivative of Equation (34) is taken with respect tothermal conductivity, which equals the average indoor minus outdoortemperatures times the number of hours divided by HVAC systemefficiency. Note that initial thermal efficiency is irrelevant in theequation. The effect of a change in thermal conductivity UA^(Total)(process 72) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = \frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}} & (35)\end{matrix}$

Change in Average Indoor Temperature

A change in average indoor temperature can also affect a change in fuelrequirements. The derivative of Equation (34) is taken with respect tothe average indoor temperature. Since UA^(Balance Point) is also afunction of average indoor temperature, application of the product ruleis required. After simplifying, the effect of a change in average indoortemperature (process 73) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\overset{\_}{T^{Indoor}}} = {\left( {UA}^{Total} \right)\left( \frac{H}{\eta^{HVAC}} \right)}} & (36)\end{matrix}$

Change in HVAC System Efficiency

As well, a change in HVAC system efficiency can affect a change in fuelrequirements. The derivative of Equation (34) is taken with respect toHVAC system efficiency, which equals current fuel consumption divided byHVAC system efficiency. Note that this term is not linear withefficiency and thus is valid for small values of efficiency changes. Theeffect of a change in fuel requirements relative to the change in HVACsystem efficiency (process 74) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\;\eta^{HVAC}} = {- {Q^{Fuel}\left( \frac{1}{\eta^{HVAC}} \right)}}} & (37)\end{matrix}$

Change in Solar Gains

An increase in solar gains can be accomplished by increasing theeffective area of south-facing windows. Effective area can be increasedby trimming trees blocking windows, removing screens, cleaning windows,replacing windows with ones that have higher SHGCs, installingadditional windows, or taking similar actions. In this case, thevariable of interest is the effective window area W. The total gain persquare meter of additional effective window area equals the availableresource (kWh/m²) divided by HVAC system efficiency, converted to Btus.The derivative of Equation (34) is taken with respect to effectivewindow area. The effect of an increase in solar gains (process 74) canbe evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\overset{\_}{W}} = {{- \left\lbrack \frac{\left( \overset{\_}{Solar} \right)(H)}{\eta^{HVAC}} \right\rbrack}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{kWh} \right)}} & (38)\end{matrix}$

Discussion

Both Equations (1) and (34) provide ways to calculate fuel consumptionrequirements. The two equations differ in several key ways:

-   -   1. UA^(Total) only occurs in one place in Equation (34), whereas        Equation (1) has multiple indirect and non-linear dependencies        to UA^(Total).    -   2. UA^(Total) is divided into two parts in Equation (34), while        there is only one occurrence of UA^(Total) in Equation (1).    -   3. The concept of balance point thermal conductivity in        Equation (34) replaces the concept of balance point temperature        in Equation (1).    -   4. Heat from occupants, electricity consumption, and solar gains        are grouped together in Equation (34) as internal heating gains,        while these values are treated separately in Equation (1).

Second, Equations (28) through (31) provide empirical methods todetermine both the point at which a building has no auxiliary heatingrequirements and the current thermal conductivity. Equation (1)typically requires a full detailed energy audit to obtain the datarequired to derive thermal conductivity. In contrast, Equations (25)through (28), as applied through the first approach, can substantiallyreduce the scope of an energy audit.

Third, both Equation (4) and Equation (35) provide ways to calculate achange in fuel requirements relative to a change in thermalconductivity. However, these two equations differ in several key ways:

-   -   1. Equation (4) is complex, while Equation (35) is simple.    -   2. Equation (4) depends upon current building thermal        conductivity, balance point temperature, solar savings fraction,        auxiliary heating efficiency, and a variety of other        derivatives. Equation (35) only requires the auxiliary heating        efficiency in terms of building-specific information.

Equation (35) implies that, as long as some fuel is required forauxiliary heating, a reasonable assumption, a change in fuelrequirements will only depend upon average indoor temperature (asapproximated by thermostat setting), average outdoor temperature, thenumber of hours (or other time units) in the (heating) season, and HVACsystem efficiency. Consequently, any building shell (or envelope)investment can be treated as an independent investment. Importantly,Equation (35) does not require specific knowledge about buildingconstruction, age, occupancy, solar gains, internal electric gains, orthe overall thermal conductivity of the building. Only thecharacteristics of the portion of the building that is being replaced,the efficiency of the HVAC system, the indoor temperature (as reflectedby the thermostat setting), the outdoor temperature (based on location),and the length of the winter season are required; knowledge about therest of the building is not required. This simplification is a powerfuland useful result.

Fourth, Equation (36) provides an approach to assessing the impact of achange in indoor temperature, and thus the effect of making a change inthermostat setting. Note that Equation (31) only depends upon theoverall efficiency of the building, that is, the building's totalthermal conductivity UA^(Total), the length of the winter season (innumber of hours or other time units), and the HVAC system efficiency;Equation (31) does not depend upon either the indoor or outdoortemperature.

Equation (31) is useful in assessing claims that are made by HVACmanagement devices, such as the Nest thermostat device, manufactured byNest Labs, Inc., Palo Alto, Calif., or the Lyric thermostat device,manufactured by Honeywell Int'l Inc., Morristown, N.J., or otherso-called “smart” thermostat devices. The fundamental idea behind thesetypes of HVAC management devices is to learn behavioral patterns, sothat consumers can effectively lower (or raise) their average indoortemperatures in the winter (or summer) months without affecting theirpersonal comfort. Here, Equation (31) could be used to estimate thevalue of heating and cooling savings, as well as to verify the consumerbehaviors implied by the new temperature settings.

Balance Point Temperature

Before leaving this section, balance point temperature should briefly bediscussed. The formulation in this first approach does not involvebalance point temperature as an input. A balance point temperature,however, can be calculated to equal the point at which there is no fuelconsumption, such that there are no gains associated with auxiliaryheating (Q^(Gains-Auxilary Heating) equals 0) and the auxiliary heatingthermal conductivity (UA^(Auxiliary Heating) in Equation (31)) is zero.Inserting these assumptions into Equation (19) and labeling T^(utdoor)as T^(Balance Point) yields:Q ^(Gains-Internal) =UA ^(Total)( T ^(Indoor) −T^(Balance Point))(H)  (39)

Equation (39) simplifies to:

$\begin{matrix}{{{\overset{\_}{T}}^{{Balance}\mspace{11mu}{Point}} = {{\overset{\_}{T}}^{Indoor} - {\frac{{\overset{\_}{Q}}^{{Gains} - {Internal}}}{{UA}^{Total}}\mspace{14mu}{where}}}}{{\overset{\_}{Q}}^{{Gains} - {Internal}} = \frac{Q^{{Gains} - {Internal}}}{H}}} & (40)\end{matrix}$

Equation (40) is identical to Equation (2), except that average valuesare used for indoor temperature T ^(Indoor), balance point temperature T^(Balance Point), and fuel consumption for internal heating gains Q^(Gains-Internal), and that heating gains from occupancy(Q^(Gains-Occupants)), electric (Q^(Gains-Electric)), and solar(Q^(Gains-Solar)) are all included as part of internal heating gains(Q^(Gains-Internal)).

Application: Change in Thermal Conductivity Associated with OneInvestment

An approach to calculating a new value for total thermal conductivity

^(Total) after a series of M changes (or investments) are made to abuilding is described in commonly-assigned U.S. patent application,entitled “System and Method for Interactively Evaluating PersonalEnergy-Related Investments,” Ser. No. 14/294,079, filed Jun. 2, 2014,pending, the disclosure of which is incorporated by reference. Theapproach is summarized therein in Equation (41), which provides:

Total = UA Total + ∑ j = 1 M ⁢ ( U j - U ^ j ) ⁢ A j + ρ ⁢ ⁢ c ⁡ ( n - n ^ ) ⁢V ( 41 )where a caret symbol (^) denotes a new value, infiltration losses arebased on the density of air (ρ), specific heat of air (c), number of airchanges per hour (n), and volume of air per air change (V). In addition,U^(j) and Û^(j) respectively represent the existing and proposedU-values of surface j, and A^(j) represents the surface area of surfacej. The volume of the building V can be approximated by multiplyingbuilding square footage by average ceiling height. The equation, with aslight restatement, equals:

Total = UA Total + Δ ⁢ ⁢ UA Total ⁢ ⁢ and ( 42 ) Δ ⁢ ⁢ UA Total = ∑ j = 1 M ⁢ (U j - U ^ j ) ⁢ A j + ρ ⁢ ⁢ c ⁡ ( n - n ^ ) ⁢ V . ( 43 )

If there is only one investment, the m superscripts can be dropped andthe change in thermal conductivity UA^(Total) equals the area (A) timesthe difference of the inverse of the old and new R-values R and{circumflex over (R)}:

$\begin{matrix}{{\Delta\;{UA}^{\;{Total}}} = {{A\left( {U - \hat{U}} \right)} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}.}}} & (44)\end{matrix}$

Fuel Savings

The fuel savings associated with a change in thermal conductivityUA^(Total) for a single investment equals Equation (44) times (35):

$\begin{matrix}{{\Delta\; Q^{Fuel}} = {{\Delta\;{UA}^{Total}} = {\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}\frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}}}}} & (45)\end{matrix}$where ΔQ^(Fuel) signifies the change in fuel consumption.

Economic Value

The economic value of the fuel savings (Annual Savings) equals the fuelsavings times the average fuel price (Price) for the building inquestion:

${{Annual}\mspace{14mu}{Savings}} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}\frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}({Price})}$where

$\begin{matrix}{{Price} = \left\{ \begin{matrix}{\frac{{Price}^{NG}}{10^{5}}{if}\mspace{14mu}{price}\mspace{14mu}{has}\mspace{14mu}{units}\mspace{14mu}{of}\mspace{14mu}\$\mspace{14mu}{per}\mspace{14mu}{therm}} \\{\frac{{Price}^{Electrity}}{3\text{,}412}{if}\mspace{14mu}{price}\mspace{14mu}{has}\mspace{14mu}{units}\mspace{14mu}{of}\mspace{14mu}\$\mspace{14mu}{per}\mspace{14mu}{kWh}}\end{matrix} \right.} & (46)\end{matrix}$where Price^(NG) represents the price of natural gas andPrice^(Electrity) represents the price of electricity. Other pricingamounts, pricing conversion factors, or pricing expressions arepossible.

Example

Consider an example. A consumer in Napa, Calif. wants to calculate theannual savings associating with replacing a 20 ft² single-pane windowthat has an R-value of 1 with a high efficiency window that has anR-value of 4. The average temperature in Napa over the 183-day winterperiod (4,392 hours) from October 1 to March 31 is 50° F. The consumersets his thermostat at 68° F., has a 60 percent efficient natural gasheating system, and pays $1 per therm for natural gas. How much moneywill the consumer save per year by making this change?

Putting this information into Equation (46) suggests that he will save$20 per year:

$\begin{matrix}{{{Annual}\mspace{14mu}{Savings}} = {{20\left( {\frac{1}{1} - \frac{1}{4}} \right)\frac{\left( {68 - 50} \right)\left( {4\text{,}392} \right)}{0.6}\left( \frac{1}{10^{5}} \right)} = {\$ 20}}} & (47)\end{matrix}$

Application: Validate Building Shell Improvements Savings

Many energy efficiency programs operated by power utilities grapple withthe issue of measurement and evaluation (M&E), particularly with respectto determining whether savings have occurred after building shellimprovements were made. Equations (28) through (31) can be applied tohelp address this issue. These equations can be used to calculate abuilding's total thermal conductivity UA^(Total). This result providesan empirical approach to validating the benefits of building shellinvestments using measured data.

Equations (28) through (31) require the following inputs:

1) Weather:

-   -   a) Average outdoor temperature (° F.).    -   b) Average indoor temperature (° F.).    -   c) Average direct solar resource on a vertical, south-facing        surface.

2) Fuel and energy:

-   -   a) Average gross indoor electricity consumption.    -   b) Average natural gas fuel consumption for space heating.    -   c) Average electric fuel consumption for space heating.

3) Other inputs:

-   -   a) Average number of occupants.    -   b) Effective window area.    -   c) HVAC system efficiency.

Weather data can be determined as follows. Indoor temperature can beassumed based on the setting of the thermostat (assuming that thethermostat's setting remained constant throughout the time period), ormeasured and recorded using a device that takes hourly or periodicindoor temperature measurements, such as a Nest thermostat device or aLyric thermostat device, cited supra, or other so-called “smart”thermostat devices. Outdoor temperature and solar resource data can beobtained from a service, such as Solar Anywhere SystemCheck, citedsupra, or the National Weather Service. Other sources of weather dataare possible.

Fuel and energy data can be determined as follows. Monthly utilitybilling records provide natural gas consumption and net electricitydata. Gross indoor electricity consumption can be calculated by addingPV production, whether simulated using, for instance, the Solar AnywhereSystemCheck service, cited supra, or measured directly, and subtractingout external electricity consumption, that is, electricity consumptionfor electric devices that do not deliver all heat that is generated intothe interior of the building. External electricity consumption includeselectric vehicle (EV) charging and electric water heating. Other typesof external electricity consumption are possible. Natural gasconsumption for heating purposes can be estimated by subtractingnon-space heating consumption, which can be estimated, for instance, byexamining summer time consumption using an approach described incommonly-assigned U.S. patent application, entitled “System and Methodfor Facilitating Implementation of Holistic Zero Net EnergyConsumption,” Ser. No. 14/531,933, filed Nov. 3, 2014, pending, thedisclosure of which is incorporated by reference. Other sources of fueland energy data are possible.

Finally, the other inputs can be determined as follows. The averagenumber of occupants can be estimated by the building owner or occupant.Effective window area can be estimated by multiplying actualsouth-facing window area times solar heat gain coefficient (estimated orbased on empirical tests, as further described infra), and HVAC systemefficiency can be estimated (by multiplying reported furnace ratingtimes either estimated or actual duct system efficiency), or can bebased on empirical tests, as further described infra. Other sources ofdata for the other inputs are possible.

Consider an example. FIG. 7 is a table 80 showing, by way of example,data used to calculate thermal conductivity. The data inputs are for asample house in Napa, Calif. based on the winter period of October 1 toMarch 31 for six winter seasons, plus results for a seventh winterseason after many building shell investments were made. (Note thebuilding improvements facilitated a substantial increase in the averageindoor temperature by preventing a major drop in temperature duringnight-time and non-occupied hours.) South-facing windows had aneffective area of 10 m² and the solar heat gain coefficient is estimatedto be 0.25 for an effective window area of 2.5 m². The measured HVACsystem efficiency of 59 percent was based on a reported furnaceefficiency of 80 percent and an energy audit-based duct efficiency of 74percent.

FIG. 8 is a table 90 showing, by way of example, thermal conductivityresults for each season using the data in the table 80 of FIG. 7 asinputs into Equations (28) through (31). Thermal conductivity is inunits of Btu/h-° F. FIG. 9 is a graph 100 showing, by way of example, aplot of the thermal conductivity results in the table 90 of FIG. 8. Thex-axis represents winter seasons for successive years, each winterseason running from October 1 to March 31. The y-axis represents thermalconductivity. The results from a detailed energy audit, performed inearly 2014, are superimposed on the graph. The energy audit determinedthat the house had a thermal conductivity of 773 Btu/h-° F. The averageresult estimated for the first six seasons was 791 Btu/h-° F. A majoramount of building shell work was performed after the 2013-2014 winterseason, and the results show a 50-percent reduction in heating energyconsumption in the 2014-2015 winter season.

Application: Evaluate Investment Alternatives

The results of this work can be used to evaluate potential investmentalternatives. FIG. 10 is a graph 110 showing, by way of example, anauxiliary heating energy analysis and energy consumption investmentoptions. The x-axis represents total thermal conductivity, UA^(Total) inunits of Btu/hr-° F. The y-axis represents total heating energy. Thegraph presents the analysis of the Napa, Calif. building from theearlier example, supra, using the equations previously discussed. Thethree lowest horizontal bands correspond to the heat provided throughinternal gains 111, including occupants, heat produced by operatingelectric devices, and solar heating. The solid circle 112 represents theinitial situation with respect to heating energy consumption. Thediagonal lines 113 a, 113 b, 113 c represent three alternative heatingsystem efficiencies versus thermal conductivity (shown in the graph asbuilding losses). The horizontal dashed line 114 represents an option toimprove the building shell and the vertical dashed line 115 representsan option to switch to electric resistance heating. The plain circle 116represents the final situation with respect to heating energyconsumption.

Other energy consumption investment options (not depicted) are possible.These options include switching to an electric heat pump, increasingsolar gain through window replacement or tree trimming (this optionwould increase the height of the area in the graph labeled “SolarGains”), or lowering the thermostat setting. These options can becompared using the approach described with reference to Equations (25)through (28) to compare the options in terms of their costs and savings,which will help the homeowner to make a wiser investment.

Second Approach: Time Series Fuel Consumption

The previous section presented an annual fuel consumption model. Thissection presents a detailed time series model. This section alsocompares results from the two methods and provides an example of how toapply the on-site empirical tests.

Building-Specific Parameters

The building temperature model used in this second approach requiresthree building parameters: (1) thermal mass; (2) thermal conductivity;and (3) effective window area. FIG. 11 is a functional block diagramshowing thermal mass, thermal conductivity, and effective window arearelative to a structure 121. By way of introduction, these parameterswill now be discussed.

Thermal Mass (M)

The heat capacity of an object equals the ratio of the amount of heatenergy transferred to the object and the resulting change in theobject's temperature. Heat capacity is also known as “thermalcapacitance” or “thermal mass” (122) when used in reference to abuilding. Thermal mass Q is a property of the mass of a building thatenables the building to store heat, thereby providing “inertia” againsttemperature fluctuations. A building gains thermal mass through the useof building materials with high specific heat capacity and high density,such as concrete, brick, and stone.

The heat capacity is assumed to be constant when the temperature rangeis sufficiently small. Mathematically, this relationship can beexpressed as:Q _(Δt) =M(T _(t+Δt) ^(Indoor) −T _(t) ^(Indoor))  (48)where M equals the thermal mass of the building and temperature units Tare in ° F. Q is typically expressed in Btu or Joules. In that case, Mhas units of Btu/° F. Q can also be divided by 1 kWh/3,412 Btu toconvert to units of kWh/° F.

Thermal Conductivity (UA^(Total))

The building's thermal conductivity UA^(Total) (123) is the amount ofheat that the building gains or losses as a result of conduction andinfiltration. Thermal conductivity UA^(Total) was discussed supra withreference to the first approach for modeling annual heating fuelconsumption.

Effective Window Area (W)

The effective window area (in units of m²) (124), also discussed indetail supra, specifies how much of an available solar resource isabsorbed by the building. Effective window area is the dominant means ofsolar gain in a typical building during the winter and includes theeffect of physical shading, window orientation, and the window's solarheat gain coefficient. In the northern hemisphere, the effective windowarea is multiplied by the available average direct irradiance on avertical, south-facing surface (kW/m²), times the amount of time (H) toresult in the kWh obtained from the windows.

Energy Gain or Loss

The amount of heat transferred to or extracted from a building (Q) overa time period of Δt is based on a number of factors, including:

-   -   1) Loss (or gain if outdoor temperature exceeds indoor        temperature) due to conduction and infiltration and the        differential between the indoor and outdoor temperatures.    -   2) Gain, when the HVAC system is in the heating mode, or loss,        when the HVAC system is in the cooling mode.    -   3) Gain associated with:        -   a) Occupancy and heat given off by people.        -   b) Heat produced by consuming electricity inside the            building.        -   c) Solar radiation.

Mathematically, Q can be expressed as:

$\begin{matrix}{Q_{\Delta\; t} = {\left\lbrack {\overset{\overset{{Evelope}\mspace{14mu}{Gain}\mspace{14mu}{or}\mspace{14mu}{Loss}}{︷}}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + \overset{\overset{{Occupancy}\mspace{14mu}{Gain}}{︷}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}\mspace{14mu}{Electric}\mspace{14mu}{Gain}}{︷}}{\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + \overset{\overset{{Solar}\mspace{14mu}{Gain}}{︷}}{W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + \overset{\overset{{HVAC}\mspace{14mu}{Gain}\mspace{14mu}{or}\mspace{14mu}{Loss}}{︷}}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}} \right\rbrack\Delta\; t}} & (49)\end{matrix}$where:

-   -   Except as noted otherwise, the bars over the variable names        represent the average value over Δt hours, that is, the duration        of the applicable empirical test. For instance, T ^(Outdoor)        represents the average outdoor temperature between the time        interval of t and t+Δt.    -   UA^(Total) is the thermal conductivity (in units of Btu/hour-°        F.).    -   W is the effective window area (in units of m²).    -   Occupancy Gain is based on the average number of people (P) in        the building during the applicable empirical test (and the heat        produced by those people). The average person is assumed to        produce 250 Btu/hour.    -   Internal Electric Gain is based on heat produced by indoor        electricity consumption (Electric), as averaged over the        applicable empirical test, but excludes electricity for purposes        that do not produce heat inside the building, for instance,        electric hot water heating where the hot water is discarded down        the drain, or where there is no heat produced inside the        building, such as is the case with EV charging.    -   Solar Gain is based on the average available normalized solar        irradiance (Solar) during the applicable empirical test (with        units of kW/m²). This value is the irradiance on a vertical        surface to estimate solar received on windows; global horizontal        irradiance (GHI) can be used as a proxy for this number when W        is allowed to change on a monthly basis.    -   HVAC Gain or Loss is based on whether the HVAC is in heating or        cooling mode (GainOrLoss is 1 for heating and −1 for cooling),        the rating of the HVAC system (R in Btu), HVAC system efficiency        (η^(HVAC), including both conversion and delivery system        efficiency), average operation status (Status) during the        empirical test, a time series value that is either off (0        percent) or on (100 percent),    -   Other conversion factors or expressions are possible.

Energy Balance

Equation (48) reflects the change in energy over a time period andequals the product of the temperature change and the building's thermalmass. Equation (49) reflects the net gain in energy over a time periodassociated with the various component sources. Equation (48) can be setto equal Equation (49), since the results of both equations equal thesame quantity and have the same units (Btu). Thus, the total heat changeof a building will equal the sum of the individual heat gain/losscomponents:

$\begin{matrix}{\overset{\overset{{Total}\mspace{14mu}{Heat}\mspace{14mu}{Change}}{︷}}{M\left( {T_{t + {\Delta\; t}}^{Indoor} - T_{t}^{Indoor}} \right)} = {\left\lbrack {\overset{\overset{{Evelope}\mspace{14mu}{Gain}\mspace{14mu}{or}\mspace{14mu}{Loss}}{︷}}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + \overset{\overset{{Occupancy}\mspace{14mu}{Gain}}{︷}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}\mspace{14mu}{Electric}\mspace{14mu}{Gain}}{︷}}{\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + \overset{\overset{{Solar}\mspace{14mu}{Gain}}{︷}}{W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + \overset{\overset{{HVAC}\mspace{14mu}{Gain}\mspace{14mu}{or}\mspace{14mu}{Loss}}{︷}}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}} \right\rbrack\Delta\; t}} & (50)\end{matrix}$

Equation (50) can be used for several purposes. FIG. 12 is a flowdiagram showing a computer-implemented method 130 for modeling intervalbuilding heating energy consumption in accordance with a furtherembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 23, as a series of process or method modules or steps.

As a single equation, Equation (50) is potentially very useful, despitehaving five unknown parameters. In this second approach, the unknownparameters are solved by performing a series of short duration empiricaltests (step 131), as further described infra with reference to FIG. 14.Once the values of the unknown parameters are found, a time series ofindoor temperature data can be constructed (step 132), which will thenallow annual fuel consumption to be calculated (step 133) and maximumindoor temperature to be found (step 134). The short duration tests willfirst be discussed.

Empirically Determine Building- and Equipment-Specific Parameters UsingShort Duration Tests

A series of tests can be used to iteratively solve Equation (50) toobtain the values of the unknown parameters by ensuring that theportions of Equation (50) with the unknown parameters are equal to zero.These tests are assumed to be performed when the HVAC is in heating modefor purposes of illustration. Other assumptions are possible.

FIG. 13 is a table 140 showing the characteristics of empirical testsused to solve for the five unknown parameters in Equation (50). Theempirical test characteristics are used in a series ofsequentially-performed short duration tests; each test builds on thefindings of earlier tests to replace unknown parameters with foundvalues.

The empirical tests require the use of several components, including acontrol for turning an HVAC system ON or OFF, depending upon the test;an electric controllable interior heat source; a monitor to measure theindoor temperature during the test; a monitor to measure the outdoortemperature during the test; and a computer or other computationaldevice to assemble the test results and finding thermal conductivity,thermal mass, effective window area, and HVAC system efficiency of abuilding based on the findings. The components can be separate units, orcould be consolidated within one or more combined units. For instance, acomputer equipped with temperature probes could both monitor, record andevaluate temperature findings. FIG. 14 is a flow diagram showing aroutine 150 for empirically determining building- and equipment-specificparameters using short duration tests for use in the method 130 of FIG.12. The approach is to run a serialized series of empirical tests. Thefirst test solves for the building's total thermal conductivity(UA^(Total)) (step 151). The second test uses the empirically-derivedvalue for UA^(Total) to solve for the building's thermal mass (M) (step152). The third test uses both of these results, thermal conductivityand thermal mass, to find the building's effective window area (W) (step153). Finally, the fourth test uses the previous three test results todetermine the overall HVAC system efficiency (step 145). Consider how toperform each of these tests.

Test 1: Building Thermal Conductivity (UA^(Total))

The first step is to find the building's total thermal conductivity(UA^(Total)) (step 151). Referring back to the table in FIG. 13, thisshort-duration test occurs at night (to avoid any solar gain) with theHVAC system off (to avoid any gain from the HVAC system), and by havingthe indoor temperature the same at the beginning and the ending of thetest by operating an electric controllable interior heat source, such asportable electric space heaters that operate at 100% efficiency, so thatthere is no change in the building temperature's at the beginning and atthe ending of the test. Thus, the interior heart source must havesufficient heating capacity to maintain the building's temperaturestate. Ideally, the indoor temperature would also remain constant toavoid any potential concerns with thermal time lags.

These assumptions are input into Equation (50):

$\begin{matrix}{{M(0)} = {\left\lbrack {{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + {(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {{W(0)}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {(1)R^{HVAC}{\eta^{HVAC}(0)}}} \right\rbrack\Delta\; t}} & (51)\end{matrix}$

The portions of Equation (51) that contain four of the five unknownparameters now reduce to zero. The result can be solved for UA^(Total):

$\begin{matrix}{{UA}^{Total} = \frac{\left\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} \right\rbrack}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (52)\end{matrix}$where T ^(Indoor) represents the average indoor temperature during theempirical test, T ^(Outdoor) represents the average outdoor temperatureduring the empirical test, P represents the average number of occupantsduring the empirical test, and Electric represents average indoorelectricity consumption during the empirical test.

Equation (52) implies that the building's thermal conductivity can bedetermined from this test based on average number of occupants, averagepower consumption, average indoor temperature, and average outdoortemperature.

Test 2: Building Thermal Mass (M)

The second step is to find the building's thermal mass (M) (step 152).This step is accomplished by constructing a test that guarantees Misspecifically non-zero since UA^(Total) is known based on the results ofthe first test. This second test is also run at night, so that there isno solar gain, which also guarantees that the starting and the endingindoor temperatures are not the same, that is, T_(t+Δt) ^(Indoor)≠T_(t)^(Indoor), respectively at the outset and conclusion of the test by notoperating the HVAC system. These assumptions are input into Equation(50) and solving yields a solution for M

$\begin{matrix}{M = {\left\lbrack \frac{{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + {(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}{\left( {T_{t + {\Delta\; t}}^{Indoor} - T_{t}^{Indoor}} \right)} \right\rbrack\Delta\; t}} & (53)\end{matrix}$where UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature during the empirical test, T^(Outdoor) represents the average outdoor temperature during theempirical test, P represents the average number of occupants during theempirical test, Electric represents average indoor electricityconsumption during the empirical test, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, T_(t) ^(Indoor) represents the starting indoor temperature,and T_(t+Δt) ^(Indoor)≠T_(t) ^(Indoor).

Test 3: Building Effective Window Area (W)

The third step to find the building's effective window area (W) (step153) requires constructing a test that guarantees that solar gain isnon-zero. This test is performed during the day with the HVAC systemturned off. Solving for W yields:

$\begin{matrix}{W = {\left\{ {\left\lbrack \frac{M\left( {T_{t + {\Delta\; t}}^{Indoor} - T_{t}^{Indoor}} \right)}{3\text{,}412\mspace{14mu}\Delta\; t} \right\rbrack - \frac{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)}{3\text{,}412} - \frac{(250)\overset{\_}{P}}{3\text{,}412} - \overset{\_}{Electric}} \right\}\left\lbrack \frac{1}{\overset{\_}{Solar}} \right\rbrack}} & (54)\end{matrix}$where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T_(t) ^(Indoor) represents the starting indoortemperature, UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature, T ^(Outdoor) represents theaverage outdoor temperature, P represents the average number ofoccupants during the empirical test, Electric represents averageelectricity consumption during the empirical test, and Solar representsthe average solar energy produced during the empirical test.

Test 4: HVAC System Efficiency (η^(Furnace) η^(Delivery)) The fourthstep determines the HVAC system efficiency (step 154). Total HVAC systemefficiency is the product of the furnace efficiency and the efficiencyof the delivery system, that is, the duct work and heat distributionsystem. While these two terms are often solved separately, the productof the two terms is most relevant to building temperature modeling. Thistest is best performed at night, so as to eliminate solar gain. Thus:

$\begin{matrix}{\eta^{HVAC} = {\left\lbrack {\frac{M\left( {T_{t + {\Delta\; t}}^{Indoor} - T_{t}^{Indoor}} \right)}{\Delta\; t} - {{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} - {(250)\overset{\_}{P}} - {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} \right\rbrack\left\lbrack \frac{1}{(1)R^{HVAC}\overset{\_}{Status}} \right\rbrack}} & (55)\end{matrix}$where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T_(t) ^(Indoor) represents the starting indoortemperature, UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature, T ^(Outdoor) represents theaverage outdoor temperature, P represents the average number ofoccupants during the empirical test, Electric represents averageelectricity consumption during the empirical test, Status represents theaverage furnace operation status, and R^(Furnace) represents the ratingof the furnace.

Note that HVAC duct efficiency can be determined without performing aduct leakage test if the generation efficiency of the furnace is known.This observation usefully provides an empirical method to measure ductefficiency without having to perform a duct leakage test.

Time Series Indoor Temperature Data

The previous subsection described how to perform a series of empiricalshort duration tests to determine the unknown parameters in Equation(50). Commonly-assigned U.S. patent application Ser. No. 14/531,933,cited supra, describes how a building's UA^(Total) can be combined withhistorical fuel consumption data to estimate the benefit of improvementsto a building. While useful, estimating the benefit requires measuredtime series fuel consumption and HVAC system efficiency data. Equation(50), though, can be used to perform the same analysis without the needfor historical fuel consumption data.

Referring back to FIG. 12, Equation (50) can be used to construct timeseries indoor temperature data (step 132) by making an approximation.Let the time period (Δt) be short (an hour or less), so that the averagevalues are approximately equal to the value at the beginning of the timeperiod, that is, assume T^(Outdoor) ≈T_(t) ^(Outdoor). The averagevalues in Equation (50) can be replaced with time-specific subscriptedvalues and solved to yield the final indoor temperature.

$\begin{matrix}{T_{t + {\Delta\; t}}^{Indoor} = {T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {{Electric}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {{WSolar}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {({HeatOrCool})R^{HVAC}\eta^{HVAC}{Status}_{t}}} \right\rbrack}\Delta\; t}}} & (56)\end{matrix}$Once T_(t+Δt) ^(Indoor) is known, Equation (56) can be used to solve forT_(t+2Δt) ^(Indoor) and so on.

Importantly, Equation (56) can be used to iteratively construct indoorbuilding temperature time series data with no specific information aboutthe building's construction, age, configuration, number of stories, andso forth. Equation (56) only requires general weather datasets (outdoortemperature and irradiance) and building-specific parameters. Thecontrol variable in Equation (56) is the fuel required to deliver theauxiliary heat at time t, as represented in the Status variable, thatis, at each time increment, a decision is made whether to run the HVACsystem.

Seasonal Fuel Consumption

Equation (50) can also be used to calculate seasonal fuel consumption(step 133) by letting Δt equal the number of hours (H) in the entireseason, either heating or cooling (and not the duration of theapplicable empirical test), rather than making Δt very short (such as anhour, as used in an applicable empirical test). The indoor temperatureat the start and the end of the season can be assumed to be the same or,alternatively, the total heat change term on the left side of theequation can be assumed to be very small and set equal to zero.Rearranging Equation (50) provides:

$\begin{matrix}{{({HeatOrCool})R^{HVAC}\eta^{HVAC}{\overset{\_}{Status}(H)}} = {{{- \left\lbrack {{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} \right\rbrack}(H)} - {\left\lbrack {{(250)P} + {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} \right\rbrack(H)}}} & (57)\end{matrix}$

Total seasonal fuel consumption based on Equation (50) can be shown tobe identical to fuel consumption calculated using the annual methodbased on Equation (34). First, Equation (57), which is a rearrangementof Equation (50), can be simplified. Multiplying Equation (57) byHeatOrCool results in (HeatOrCool)² on the left hand side, which equals1 for both heating and cooling seasons, and can thus be dropped from theequation. In addition, the sign on the first term on the right hand sideof Equation (57) ([UA^(Total)(T ^(Outdoor)−T ^(Indoor))](H)) can bechanged by reversing the order of the temperatures. Per Equation (8),the second term on the right hand side of the equation

$\left( {\left\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} \right\rbrack(H)} \right)$equals internal gains (Q^(Gains-Internal)), which can be substitutedinto Equation (57). Finally, dividing the equation by HVAC efficiencyη^(HVAC) yields:

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = {\left\lbrack {{({HeatOrCool})\left( {UA}^{Total} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)} - {({HeatOrCool})Q^{{Gains} - {Internal}}}} \right\rbrack\left( \frac{1}{\eta^{HVAC}} \right)}} & (58)\end{matrix}$Next, substituting Equation (23) into Equation (58):

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = \left\lbrack {\left. \quad{{({HeatOrCool})\left( {UA}^{Total} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)} - {({HeatOrCool})({HeatOrCool})\left( {UA}^{{Balance}\mspace{11mu}{Point}} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} \right\rbrack\left( \frac{1}{\eta^{HVAC}} \right)} \right.} & (59)\end{matrix}$Once again, HeatOrCool² equals 1 for both heating and cooling seasonsand thus is dropped. Equation (59) simplifies as:

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = \frac{\begin{matrix}\left\lbrack {{{HeatOrCool}\left( {UA}^{Total} \right)} - \left( {UA}^{{Balance}\mspace{11mu}{Point}} \right)} \right\rbrack \\{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}\end{matrix}}{\eta^{HVAC}}} & (60)\end{matrix}$

Consider the heating season when HeatOrCool equals 1. Equation (60)simplifies as follows.

$\begin{matrix}{Q^{Fuel} = \frac{\left( {{UA}^{Total} - {UA}^{{Balance}\mspace{11mu}{Point}}} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}} & (61)\end{matrix}$

Equation (61) illustrates total seasonal fuel consumption based onEquation (50) is identical to fuel consumption calculated using theannual method based on Equation (34).

Consider the cooling season when HeatOrCool equals −1. Multiply Equation(61) by the first part of the right hand side by −1 and reverse thetemperatures, substitute −1 for HeatOrCool, and simplify:

$\begin{matrix}{Q^{Fuel} = \frac{\left( {{UA}^{Total} + {UA}^{{Balance}\mspace{11mu}{Point}}} \right)\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)(H)}{\eta^{HVAC}}} & (62)\end{matrix}$

A comparison of Equations (61) and (62) shows that a leverage effectoccurs that depends upon whether the season is for heating or cooling.Fuel requirements are decreased in the heating season because internalgains cover a portion of building losses (Equation (61)). Fuelrequirements are increased in the cooling season because cooling needsto be provided for both the building's temperature gains and theinternal gains (Equation (62)).

Maximum Indoor Temperature

Allowing consumers to limit the maximum indoor temperature to some valuecan be useful from a personal physical comfort perspective. The limit ofmaximum indoor temperature (step 134) can be obtained by taking theminimum of T_(t+Δt) ^(Indoor) and T^(Indoor-Max), the maximum indoortemperature recorded for the building during the heating season. Therecan be some divergence between the annual and detailed time seriesmethods when the thermal mass of the building is unable to absorb excessheat, which can then be used at a later time. Equation (56) becomesEquation (63) when the minimum is applied.

$\begin{matrix}{T_{t + {\Delta\; t}}^{Indoor} = {{Min}\left\{ {T^{{Indoor} - {Max}},{T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {{Electric}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {W\mspace{11mu}{{Solar}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + {({HeatOrCool})R^{HVAC}\eta^{HVAC}{Status}_{t}}} \right\rbrack}\Delta\; t}}} \right\}}} & (63)\end{matrix}$

Net Fuel Savings, Net Cost Savings and Net Environmental Savings

Equation (58), which is a simplification of Equation (57), can be usedto calculate net savings in fuel, cost, and carbon emissions(environmental). Net savings are crucial, albeit frequently overlooked,factors to consider when contemplating or evaluating changes to electricenergy efficiency investments and renewable distributed powergeneration. Envelope gains or losses on the right hand side of Equation(58) are represented by the first group of terms,(HeatOrCool)(UA^(Total))(T ^(Indoor)−T ^(Outdoor))(H). Envelope gainsoccur in the summer when average indoor temperature is less averageoutdoor temperature (T ^(Indoor)<T ^(Outdoor)), while envelope lossesoccur in the winter when average indoor temperature exceeds averageoutdoor temperature (T ^(Indoor)>T ^(Outdoor)). Internal gains arerepresented by the second group of terms in the equation,(HeatOrCool)Q^(Gains-Internal), and include heat from occupants,internal electric, and solar. Thus, Equation (58) can be annotated as:

$\begin{matrix}{{\overset{{HVAC}\mspace{11mu}{Fuel}}{\overset{︷}{Q^{Fuel}}} = {\frac{\begin{matrix}{{({HeatOrCool})\overset{{Envelop}\mspace{11mu}{Gains}\;{({Losses})}}{\overset{︷}{\left( {UA}^{Total} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}}} -} \\{({HeatOrCool})\overset{\begin{matrix}\begin{matrix}{{Occupancy},} \\{{Internal}\mspace{14mu}{Electric}}\end{matrix} \\{{and}\mspace{14mu}{Solar}\mspace{14mu}{Gains}}\end{matrix}}{\overset{︷}{\left( Q^{{Internal}\mspace{11mu}{Gains}} \right)}}}\end{matrix}}{\eta^{HVAC}}\mspace{14mu}{where}}}{{HeatOrCool} = \left\{ {\begin{matrix}1 & {{for}\mspace{14mu}{Heating}\mspace{14mu}{Season}} \\{- 1} & {{for}\mspace{14mu}{Cooling}\mspace{14mu}{Season}}\end{matrix}.} \right.}} & (64)\end{matrix}$

As an initial step, envelope gains can be multiplied by the binary termHeatOrCool to produce a term that will be a positive number across bothseasons. More specifically, average indoor temperature is assumed to beless than the average outdoor temperature in the summer, so that thedifference between the two temperatures will generally be a negativenumber. The resulting term for envelope gains becomes a positive numberwhen multiplied by the binary term HeatOrCool for summer, −1. Averageoutdoor temperature is assumed to be less than the average indoortemperature in the winter, so envelope gains will generally be apositive number. Internal gains are also multiplied by the binary termHeatOrCool, but for a different reason. Internal gains in the winterreduce the need for auxiliary heating because these types of heat gains,heating gained from occupants Q^(Gains-Occupants) heating gained fromthe operation of electric devices Q^(Gains-Electric), and heating gainedfrom solar heating Q^(Gains-Solar), provide some of the heating to thestructure. Internal gains in the summer increase the need for auxiliarycooling, that is, the cooling system needs to provide enough cooling forboth the envelope gains and the internal gains.

Definition of UA^(Balance Point)

Assume there exists a term called UA^(Balance Point) that satisfies thefollowing relationship, where the binary term HeatOrCool is included tokeep UA^(Balance Point) positive across both seasons, as explainedsupra:Q ^(Internal Gains)=(HeatOrCool)(UA ^(Balance Point))( T ^(Indoor) −T^(Outdoor))(H)  (65)Solve Equation (65) for UA^(Balance Point) by dividing by (T ^(Indoor)−T^(Outdoor))(H) and multiplying by HeatOrCool, assuming T ^(Indoor)≠T^(Outdoor) and H≠0. As HeatOrCool² always equals 1, the termUA^(Balance Point) can be expressed as:

$\begin{matrix}{{UA}^{{Balance}\mspace{11mu}{Point}} = \frac{({HeatOrCool})\left( Q^{{Internal}\mspace{11mu}{Gains}} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (66)\end{matrix}$

Change in Fuel Relative to Change in Internal Gains

Consider the effect of changes in fuel relative to a change in internalgains. Substitute Equation (65) into Equation (64) and simplify:

$\begin{matrix}{Q^{Fuel} = \frac{\begin{matrix}\left\lbrack {{({HeatOrCool})\left( {UA}^{Total} \right)} -} \right. \\{\left. \left( {UA}^{{Balance}\mspace{11mu}{Point}} \right) \right\rbrack\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}\end{matrix}}{\eta^{HVAC}}} & (67)\end{matrix}$Take the derivate of Equation (64) with respect to internal gains:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dQ}^{{Internal}\mspace{11mu}{Gains}}} = {- \frac{HeatOrCool}{\eta^{HVAC}}}} & (68)\end{matrix}$

Suppose that an investment in electric energy efficiency directlyreduces electricity consumption by the amount Q^(Energy Efficeny).Reducing electricity consumption has a direct effect and two indirecteffects. The direct effect of reducing electricity consumption is thatthe fuel required to generate the electricity is reduced. The amount offuel reduced equals Q^(Energy Effciency) divided by the efficiency ofgeneration (η^(Electricity Generation)). The indirect effects are due tothe observation that the reduction in electricity consumption reducesthe amount of waste heat in the building. The lost heat needs to bereplaced in the heating season and equals Q^(Energy Efficiency) timesEquation (68) times the fraction of the year that the heating system isin operation. The lost heat reduces the burden on the HVAC system in thecooling season and equals Q^(Energy Efficency) times Equation (68) timesthe fraction of the year that the cooling system is in operation.

The net fuel savings, taking into account the effect on heating andcooling, is:

$\begin{matrix}{{{Net}\mspace{14mu}{Fuel}\mspace{14mu}{Savings}} = {\frac{Q^{{Energy}\mspace{11mu}{Efficiency}}}{\eta^{{Electricity}\mspace{11mu}{Generation}}} - \frac{Q^{{Energy}\mspace{11mu}{Efficiency}}(F)}{\eta^{{HVAC} - {Heating}}} + \frac{Q^{{Energy}\mspace{11mu}{Efficiency}}\left( {1 - F} \right)}{\eta^{{HVAC} - {Cooling}}}}} & (69)\end{matrix}$where term F represents the percent of hours that are winter hours andη^(Electricity Generation), η^(HVAC-Heating) and η^(HVAC-Cooling)respectively represent efficiencies of electricity generation assupplied to a building and of the efficiencies of the building's HVACcooling and heating systems. Other units of time could be used in placeof hours, so long as consistent with the other terms in the equation.For instance, where applicable, electricity costs are generallyexpressed in kilowatt hours (kWh), which would need to be converted intoan equivalent unit if the term F represents a time unit other thanhours. Note also that the term F models the relationship between therespective durations of the heating and cooling seasons that typicallyseasonally affect a building. Thus, the term F could similarly representthe percentage of hours (or other units of time) that are summer hours.Equation (69) simplifies to:

$\begin{matrix}{{{Net}\mspace{14mu}{Fuel}\mspace{14mu}{Savings}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left\lbrack {\frac{1}{\eta^{{Electricity}\mspace{11mu}{Generation}}} - \frac{F}{\eta^{{HVAC} - {Heating}}} + \frac{\left( {1 - F} \right)}{\eta^{{HVAC} - {Cooling}}}} \right\rbrack}} & (70)\end{matrix}$

Next, consider the net economic savings. Assume that space heating usesnatural gas and cooling uses electricity. The net economic savings is:

$\begin{matrix}{{{Net}\mspace{14mu}{Cost}\mspace{14mu}{Savings}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left\lbrack {P^{Electricity} - \frac{P^{{Natural}\mspace{11mu}{Gas}}(F)}{\eta^{{HVAC} - {Heating}}} + \frac{P^{Electricity}\left( {1 - F} \right)}{\eta^{{HVAC} - {Cooling}}}} \right\rbrack}} & (71)\end{matrix}$where P^(Electricity) and P^(Natural Gas) respectively represent theprices of electricity and natural gas. Equation (71) simplifies to:

$\begin{matrix}{{{Net}\mspace{14mu}{Cost}\mspace{14mu}{Savings}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left\{ {{P^{Electricity}\left\lbrack {1 + \frac{\left( {1 - F} \right)}{\eta^{{HVAC} - {Cooling}}}} \right\rbrack} - \frac{P^{{Natural}\mspace{11mu}{Gas}}(F)}{\eta^{{HVAC} - {Heating}}}} \right\}}} & (72)\end{matrix}$

Finally, consider carbon emissions. The net carbon emissions(environmental) savings is:

$\begin{matrix}{{{Net}\mspace{14mu}{Carbon}\mspace{14mu}{Savings}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left\{ {{E^{Electricity}\left\lbrack {1 + \frac{\left( {1 - F} \right)}{\eta^{{HVAC} - {Cooling}}}} \right\rbrack} - \frac{E^{{Natural}\mspace{11mu}{Gas}}(F)}{\eta^{{HVAC} - {Heating}}}} \right\}}} & (73)\end{matrix}$

Example

By way of illustration, net fuel savings, net cost savings, and netcarbon emissions (environmental) savings will be calculated based on thefollowing assumptions for a building that is heated with an HVAC systemthat uses natural gas to generate heating and electricity to generatecooling:

-   -   6-month heating and 6-month cooling seasons.    -   Electricity generation efficiency is 50%.    -   Heating efficiency is 65%.    -   Cooling efficiency is 400% (that is, 13.6 SEER).    -   Electricity prices is $0.17/kWh.    -   Natural gas price is $1/therm (or $0.034/kWh).    -   Electricity carbon emissions are 0.73 lbs/kWh    -   Natural gas carbon emissions are 11.7 lbs/therm (or 0.40        lbs/kWh)

First, net fuel savings is determined:

$\begin{matrix}{{{Net}\mspace{14mu}{Fuel}\mspace{14mu}{Savings}} = {{Q^{{Energy}\mspace{11mu}{Efficiency}}\left\lbrack {\frac{1}{0.5} - \frac{0.5}{0.65} + \frac{0.5}{4.00}} \right\rbrack} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left( {136\%} \right)}}} & (74)\end{matrix}$

Direct fuel savings equals 200% of energy efficiency savings sinceelectricity generation is 50% efficient. Net fuel savings, though, equalonly 136% of electricity savings, which is about two-thirds of thedirect fuel savings and does not even include the net effects ofbuilding heating and cooling that could further change net fuel savings.

Next, net cost savings is determined:

$\begin{matrix}{{{Net}\mspace{14mu}{Cost}\mspace{14mu}{Savings}} = {{Q^{{Energy}\mspace{11mu}{Efficiency}}\left\{ {{(0.17)\left\lbrack {1 + \frac{0.5}{4.0}} \right\rbrack} - \frac{(0.034)(0.5)}{0.65}} \right\}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left( {{\$ 0}{{.165}/{kWh}}} \right)}}} & (75)\end{matrix}$

Net cost savings are only slightly less than direct cost savings at$0.17/kWh. Finally, net environmental savings is determined:

$\begin{matrix}{{{Net}\mspace{14mu}{Carbon}\mspace{14mu}{Savings}} = {{Q^{{Energy}\mspace{11mu}{Efficiency}}\left\{ {{0.73{\frac{lbs}{kWh}\left\lbrack {1 + \frac{0.5}{4.00}} \right\rbrack}} - \frac{\left( {0.40\frac{lbs}{kWh}} \right)(0.5)}{0.65}} \right\}} = {Q^{{Energy}\mspace{11mu}{Efficiency}}\left\lbrack {0.51\frac{lbs}{kWh}} \right\rbrack}}} & (76)\end{matrix}$

Actual carbon savings equal 0.51 lbs/kWh, rather than 0.73 lbs/kWh,which is only 70 percent of the direct result.

This example illustrates how direct savings may be less impactful whentaken in light of net savings. The net savings in fuel, cost and carbonemissions (environmental), as respectively calculated using Equations(71), (72) and (73), enable the full effects that electric energyefficiency investments have on reductions in the fuel consumed for abuilding's heating and cooling to be weighed. The net savings realizedmay actually be less than what would seem an intuitive result. Forinstance, switching from inefficient incandescent light bulbs lowersindoor heat gain during the winter, yet more natural gas needs to beconsumed to make up for the indirect heating previously provided bythose light bulbs. Similarly, while natural gas is less expensive thanelectricity, the savings in carbon emissions may not be realized to thesame extent. In addition, renewable distributed generation has at timesbeen considered as having the same effect as energy efficiency, which isactually not the case. 1 kWh of PV power generation would need 1.5 kWhof energy efficiency savings to have the same energy or carbon emissionssavings result in the case described supra

Generalization of Energy Balance Equation

Equation (50) provides that the total heat change over a given period oftime equals the sum of the heat gain (or loss) due to envelope gains (orlosses), occupancy gains, internal electric gains, solar gains, andauxiliary gains. Equation (50) be rearranged so as to create a heatbalance equation that is composed of heat gain (loss) from six sources,as shown in Table 1. Three of the sources can contribute to either heatgain or loss, while the remaining three sources can only contribute toheat gain.

$\begin{matrix}{0 = {{\left\lbrack {\overset{{Envelope}\mspace{14mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)}} + \overset{{Occupancy}\mspace{14mu}{Gain}}{\overset{︷}{(250)\overset{\_}{P}}} + \overset{{Internal}\mspace{11mu}{Electric}\mspace{11mu}{Gain}}{\overset{︷}{\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + \overset{{Solar}\mspace{14mu}{Gain}}{\overset{︷}{W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} + \overset{{HVAC}\mspace{11mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}}} \right\rbrack\Delta\; t} + \overset{{Thermal}\mspace{14mu}{Mass}\mspace{14mu}{Gain}\mspace{14mu}{({Loss})}}{\overset{︷}{M\left( {T_{t}^{Indoor} - T_{t + {\Delta\; t}}^{Indoor}} \right)}}}} & (77)\end{matrix}$

TABLE 1 Provides Heat Provides Heat Source Gain Loss Envelope ✓ ✓ HVAC ✓✓ Thermal mass ✓ ✓ Occupancy ✓ Internal electric ✓ Solar ✓

Existing Model of Envelope Gain or Loss

Heat gain (loss) from the envelope gain (or loss) is modeled to be thebuilding's total thermal conductivity UA^(Total) times the differencebetween the average outdoor and indoor temperatures times the timeperiod Δt:Q ^(Envelope Gain(Loss)) =UA ^(Total)( T ^(Outdoor)- T^(Indoor))Δt  (78)

Equation (78) assumes the differential between outdoor and indoortemperatures is the same for all building surfaces. This assumptionmeans that the sum of the thermal conductivity UA^(Total) for allsurfaces times the temperature differential equals the total thermalconductivity times the temperature differential, that is, Σ_(surface)UA^(surface) (T ^(Outdoor)−T ^(Indoor))=UA^(Total)(T ^(Outdoor)−T^(Indoor)). Thus, the building's envelope gains and losses can bemodeled using a single thermal conductivity term UA^(Total)=Σ_(surface)UA^(surface).

While this simplifying assumption is acceptable in the winter, the sameassumption may be too simplistic for the summer. In particular, atticstend to suffer significant heat buildup during the summer and thesurface temperature above the ceiling of a house may be significantlyhigher than the outdoor ambient temperature. Thus, the temperaturedifferential between the attic and (non-attic space) indoor spaces mayactually be greater than the temperature differential between theoutdoor and (non-attic space) indoor temperatures. The same observationwith respect to the attic applies to other surfaces or portions of abuilding heated by the sun.

Envelope Gain (Loss) with Different Temperature Differentials

The simplifying assumption can be addressed by first proposing amethodology using only two surfaces, which can then be generalizable toany number of surfaces, that is, made applicable to all solar heatedsurfaces and not just attics. First, divide the building's thermalconductivity into two parts, one part for the attic and the other partfor the rest of the building. The envelope gain (loss) will equal thesum of the attic component plus the rest of the house:Q ^(Envelope Gain(Loss))=[UA ^(Attic)( T ^(Attic) −T ^(Indoor))Δt+(UA^(Total) −UA ^(Attic))( T ^(Outdoor) −T ^(Indoor)]Δt  (79)Equation (79) can be rearranged as follows:Q ^(Envelope Gain(Loss))=[UA ^(Total)( T ^(Outdoor) −T ^(Indoor))+UA^(Attic) ΔT ^(Attic Over Outdoor)]Δt  (80)where ΔT ^(Attic Over Outdoor)=T ^(Attic)−T ^(Outdoor).

Effective Window Area Term for Attic

Increased attic temperatures are caused by an increase in solarradiation. Assume that the rise in attic temperature over outdoortemperature is proportional to the available solar radiation (units ofkW/m²):

$\begin{matrix}{{\overset{\_}{\Delta\; T}}^{{Attic}\mspace{11mu}{Over}\mspace{11mu}{Outdoor}} = {\alpha\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} & (81)\end{matrix}$where the constant α has units of hr-° F.-m²/Btu. Furthermore, express αas a pair of constants, UA^(Attic) in units of Btu/hour-° F., which isassumed to be known, and W^(Attic) in units of m², which needs to bedetermined:

$\begin{matrix}{\alpha = \frac{W^{Attic}}{{UA}^{Attic}}} & (82)\end{matrix}$Substitute Equations (82) and (81) into Equation (80). The UA^(Attic)terms cancel and the envelope gain (or losses) are similar to theenvelope gain (or losses) in Equation (78) with the addition of a newterm that incorporates W^(Attic) and the amount of solar radiation:

$\begin{matrix}{Q^{{Envelope}\mspace{14mu}{Gain}\mspace{11mu}{({Loss})}} = {\left\lbrack {{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + {W^{Attic}{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} \right\rbrack\Delta\; t}} & (83)\end{matrix}$

Revised Equation

Substituting the Envelope Gain (Loss) term in Equation (77) withEquation (83) yields:

$\begin{matrix}{0 = {{\left\lbrack {\overset{{Envelope}\mspace{14mu}{Gain}\mspace{14mu}{({Loss})}}{\overset{︷}{{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)} + {W^{Attic}\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} + \overset{{Occupancy}\mspace{14mu}{Gain}}{\overset{︷}{(250)\overset{\_}{P}}} + \overset{{Internal}\mspace{14mu}{Electric}\mspace{14mu}{Gain}}{\overset{︷}{\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + \overset{{Solar}\mspace{11mu}{Gain}}{\overset{︷}{W\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} + \overset{{HVAC}\mspace{14mu}{Gain}\mspace{14mu}{({Loss})}}{\overset{︷}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}}} \right\rbrack\Delta\; t} + \overset{{Thermal}\mspace{14mu}{Mass}\mspace{11mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{M\left( {T_{t}^{Indoor} - T_{t + {\Delta\; t}}^{Indoor}} \right)}}}} & (84)\end{matrix}$

Notice that the Envelope Gain (Loss) and Solar Gain terms both have afactor that depends upon the solar resource. The form of the dependenceis identical, as both of these terms include an “Effective Window Area”term. Equation (84) can be simplified by collecting like terms, suchthat:

$\begin{matrix}{0 = {{\left\lbrack {\overset{{Envelope}\mspace{11mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} \right)}} + \overset{{Occupancy}\mspace{14mu}{Gain}}{\overset{︷}{(250)\overset{\_}{P}}} + \overset{\overset{{Internal}\mspace{14mu}{Electric}\mspace{11mu}{Gain}}{︷}}{\overset{\_}{Electric}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + \overset{{Solar}\mspace{11mu}{Gain}}{\overset{︷}{\hat{W}\;{\overset{\_}{Solar}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}}} + \overset{{HVAC}\mspace{11mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}}} \right\rbrack\Delta\; t} + \overset{{Thermal}\mspace{11mu}{Mass}\mspace{11mu}{Gain}\mspace{11mu}{({Loss})}}{\overset{︷}{M\left( {T_{t}^{Indoor} - T_{t + {\Delta\; t}}^{Indoor}} \right)}}}} & (85)\end{matrix}$where W=w+W^(Attic).

Discussion

Equation (85) differs from Equation (77) only in that the EffectiveWindow Area W includes the effect of solar heat gain directly throughwindows and the effect of an increased attic temperature. Thisobservation can be extended to cover heat gain from any other surface,so long as the heat gain is assumed to be proportional to the solarirradiation. Thus, the Effective Window Area W can be interpreted morecomprehensively than simply assuming that the effective window areareflects the actual physical window area times a solar heat gaincoefficient. Rather, the Effective Window Area W signifies that acertain portion of the solar radiation enters the building throughopaque surfaces that can be thought of as having “Effective WindowAreas.” This situation occurs in portions of a building where thetemperature of the surface is greater than the outside temperature, suchas in an attic.

Parameter Specification

Implementing Equation (85) requires weather data and building-specificparameters, plus the thermal conductivity of the attic (UA^(Attic)) andthe effective window area of the attic (W^(Attic)), which can both beempirically determined through the short duration tests discussed suprawith reference to FIG. 14, or by other means. No modifications arerequired for these tests since the attic's thermal conductivity cancelsout of the equation and the attic's Effective Window Area is directlycombined with the existing Effective Window Area. Moreover, since thesetests are empirically based, the tests already account for theadditional heat gain associated with the elevated attic temperature andother surface temperatures. Thus, when the parameters are empiricallybased, the previous approach does not need to be modified to beapplicable to summer conditions.

Validation

Equation (85) was validated using data measured from an inefficient,125-year old Victorian house located in Napa, Calif. The house wascooled by two 5-kW AC units. Two temperature monitoring devices wereplaced upstairs, one monitor was placed downstairs, and one monitor wasplaced outside. The average indoor temperature was determined by firstaveraging the two upstairs temperatures and combining the result withthe downstairs temperature. Fifteen-minute electricity consumption datawas evaluated to determine when the AC units cycled ON and OFF. Theparameters of Equation (85) were derived using the approach discussedsupra with reference to FIG. 14. Indoor temperature data was thenpredicted by combining the parameters with half-hour electricityconsumption, solar radiation, and outdoor ambient temperature data. FIG.15 is a graph 155 showing, by way of example, measured and modeledhalf-hour indoor temperature from June 15 to Jul. 30, 2015. The x-axisis the date. The y-axis is the temperature measured in Fahrenheit. Thedata depicted in the graph suggests that the modeled indoor temperaturefairly reflects the measured temperature with difference between the twotemperatures being due to the operation of a whole house fan.

Comparison to Annual Method (First Approach)

Two different approaches to calculating annual fuel consumption aredescribed herein. The first approach, per Equation (34), is asingle-line equation that requires six inputs. The second approach, perEquation (63), constructs a time series dataset of indoor temperatureand HVAC system status. The second approach considers all of theparameters that are indirectly incorporated into the first approach. Thesecond approach also includes the building's thermal mass and thespecified maximum indoor temperature, and requires hourly time seriesdata for the following variables: outdoor temperature, solar resource,internal electricity consumption, and occupancy.

Both approaches were applied to the exemplary case, discussed supra, forthe sample house in Napa, Calif. Thermal mass was 13,648 Btu/° F. andthe maximum temperature was set at 72° F. The auxiliary heating energyrequirements predicted by the two approaches was then compared. FIG. 16is a graph 160 showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach. The x-axis represents total thermal conductivity,UA^(Total) in units of Btu/hr-° F. They-axis represents total heatingenergy. FIG. 16 uses the same format as the graph in FIG. 10 by applyinga range of scenarios. The red line in the graph corresponds to theresults of the hourly method. The dashed black line in the graphcorresponds to the annual method. The graph suggests that results areessentially identical, except when the building losses are very low andsome of the internal gains are lost due to house overheating, which isprevented in the hourly method, but not in the annual method.

The analysis was repeated using a range of scenarios with similarresults. FIG. 17 is a graph 170 showing, by way of example, a comparisonof auxiliary heating energy requirements with the allowable indoortemperature limited to 2° F. above desired temperature of 68° F. Here,the only cases that found any meaningful divergence occurred when themaximum house temperature was very close to the desired indoortemperature. FIG. 18 is a graph 180 showing, by way of example, acomparison of auxiliary heating energy requirements with the size ofeffective window area tripled from 2.5 m² to 7.5 m². Here, internalgains were large by tripling solar gains and there was insufficientthermal mass to provide storage capacity to retain the gains.

The conclusion is that both approaches yield essentially identicalresults, except for cases when the house has inadequate thermal mass toretain internal gains (occupancy, electric, and solar).

Example

How to perform the tests described supra using measured data can beillustrated through an example. These tests were performed between 9 PMon Jan. 29, 2015 to 6 AM on Jan. 31, 2015 on a 35 year-old, 3,000 ft²house in Napa, Calif. This time period was selected to show that all ofthe tests could be performed in less than a day-and-a-half. In addition,the difference between indoor and outdoor temperatures was not extreme,making for a more challenging situation to accurately perform the tests.

FIG. 19 is a table 190 showing, by way of example, test data. The subcolumns listed under “Data” present measured hourly indoor and outdoortemperatures, direct irradiance on a vertical south-facing surface(VDI), electricity consumption that resulted in indoor heat, and averageoccupancy. Electric space heaters were used to heat the house and theHVAC system was not operated. The first three short-duration tests,described supra, were applied to this data. The specific data used arehighlighted in gray. FIG. 20 is a table 200 showing, by way of example,the statistics performed on the data in the table 190 of FIG. 19required to calculate the three test parameters. UA^(Total) wascalculated using the data in the table of FIG. 10 and Equation (52).Thermal Mass (M) was calculated using UA^(Total), the data in the tableof FIG. 10, and Equation (53). Effective Window Area (W) was calculatedusing UA^(Total), M, the data in the table of FIG. 10, and Equation(54).

These test parameters, plus a furnace rating of 100,000 Btu/hour andassumed efficiency of 56%, can be used to generate the end-of-periodindoor temperature by substituting them into Equation (56) to yield:

$\begin{matrix}{T_{t + {\Delta\; t}}^{Indoor} = {T_{t}^{Indoor} + {{\left\lbrack \frac{1}{18\text{,}084} \right\rbrack\left\lbrack {{429\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)\mspace{11mu} P_{t}} + {3412\mspace{14mu}{Electric}_{t}} + {11\text{,}600\mspace{11mu}{Solar}_{t}} + {(1)\left( {100\text{,}000} \right)(0.56){Status}_{t}}} \right\rbrack}\Delta\; t}}} & (86)\end{matrix}$

Indoor temperatures were simulated using Equation (86) and the requiredmeasured time series input datasets. Indoor temperature was measuredfrom Dec. 15, 2014 to Jan. 31, 2015 for the test location in Napa,Calif. The temperatures were measured every minute on the first andsecond floors of the middle of the house and averaged. FIG. 21 is agraph 210 showing, by way of example, hourly indoor (measured andsimulated) and outdoor (measured) temperatures. FIG. 22 is a graph 220showing, by way of example, simulated versus measured hourly temperaturedelta (indoor minus outdoor). FIG. 21 and FIG. 22 suggest that thecalibrated model is a good representation of actual temperatures.

Energy Consumption Modeling System

Modeling energy consumption for heating (or cooling) on an annual (orperiodic) basis, as described supra with reference FIG. 3, and on anhourly (or interval) basis, as described supra beginning with referenceto FIG. 12, can be performed with the assistance of a computer, orthrough the use of hardware tailored to the purpose. FIG. 23 is a blockdiagram showing a computer-implemented system 230 for modeling buildingheating energy consumption in accordance with one embodiment. A computersystem 231, such as a personal, notebook, or tablet computer, as well asa smartphone or programmable mobile device, can be programmed to executesoftware programs 232 that operate autonomously or under user control,as provided through user interfacing means, such as a monitor, keyboard,and mouse. The computer system 231 includes hardware componentsconventionally found in a general purpose programmable computing device,such as a central processing unit, memory, input/output ports, networkinterface, and non-volatile storage, and execute the software programs232, as structured into routines, functions, and modules. In addition,other configurations of computational resources, whether provided as adedicated system or arranged in client-server or peer-to-peertopologies, and including unitary or distributed processing,communications, storage, and user interfacing, are possible.

In one embodiment, to perform the first approach, the computer system231 needs data on heating losses and heating gains, with the latterseparated into internal heating gains (occupant, electric, and solar)and auxiliary heating gains. The computer system 231 may be remotelyinterfaced with a server 240 operated by a power utility or otherutility service provider 241 over a wide area network 239, such as theInternet, from which fuel purchase data 242 can be retrieved.Optionally, the computer system 231 may also monitor electricity 234 andother metered fuel consumption, where the meter is able to externallyinterface to a remote machine, as well as monitor on-site powergeneration, such as generated by a photovoltaic system 235. Themonitored fuel consumption and power generation data can be used tocreate the electricity and heating fuel consumption data and historicalsolar resource and weather data. The computer system 231 then executes asoftware program 232 to determine annual (or periodic) heating fuelconsumption 244 based on the empirical approach described supra withreference to FIG. 3.

In a further embodiment, to assist with the empirical tests performed inthe second approach, the computer system 231 can be remotely interfacedto a heating source 236 and a thermometer 237 inside a building 233 thatis being analytically evaluated for thermal performance, thermal mass,effective window area, and HVAC system efficiency. In a furtherembodiment, the computer system 231 also remotely interfaces to athermometer 238 outside the building 163, or to a remote data sourcethat can provide the outdoor temperature. The computer system 231 cancontrol the heating source 236 and read temperature measurements fromthe thermometer 237 throughout the short-duration empirical tests. In afurther embodiment, a cooling source (not shown) can be used in place ofor in addition to the heating source 236. The computer system 231 thenexecutes a software program 232 to determine hourly (or interval)heating fuel consumption 244 based on the empirical approach describedsupra with reference to FIG. 12.

Applications

The two approaches to estimating energy consumption for heating (orcooling), hourly and annual, provide a powerful set of tools that can beused in various applications. A non-exhaustive list of potentialapplications will now be discussed. Still other potential applicationsare possible.

Application to Homeowners

Both of the approaches, annual (or periodic) and hourly (or interval),reformulate fundamental building heating (and cooling) analysis in amanner that can divide a building's thermal conductivity into two parts,one part associated with the balance point resulting from internal gainsand one part associated with auxiliary heating requirements. These twoparts provide that:

-   -   Consumers can compare their house to their neighbors' houses on        both a total thermal conductivity UA^(Total) basis and on a        balance point per square foot basis. These two numbers, total        thermal conductivity UA^(Total) and balance point per square        foot, can characterize how well their house is doing compared to        their neighbors' houses. The comparison could also be performed        on a neighborhood- or city-wide basis, or between comparably        built houses in a subdivision. Other types of comparisons are        possible.    -   As strongly implied by the empirical analyses discussed supra,        heater size can be significantly reduced as the interior        temperature of a house approaches its balance point temperature.        While useful from a capital cost perspective, a heater that was        sized based on this implication may be slow to heat up the house        and could require long lead times to anticipate heating needs.        Temperature and solar forecasts can be used to operate the        heater by application of the two approaches described supra, so        as to optimize operation and minimize consumption. For example,        if the building owner or occupant knew that the sun was going to        start adding a lot of heat to the building in a few hours, he        may choose to not have the heater turn on. Alternatively, if the        consumer was using a heater with a low power rating, he would        know when to turn the heater off to achieve desired preferences.

Application to Building Shell Investment Valuation

The economic value of heating (and cooling) energy savings associatedwith any building shell improvement in any building has been shown to beindependent of building type, age, occupancy, efficiency level, usagetype, amount of internal electric gains, or amount solar gains, providedthat fuel has been consumed at some point for auxiliary heating. Asindicated by Equation (46), the only information required to calculatesavings includes the number of hours that define the winter season;average indoor temperature; average outdoor temperature; the building'sHVAC system efficiency (or coefficient of performance for heat pumpsystems); the area of the existing portion of the building to beupgraded; the R-value of the new and existing materials; and the averageprice of energy, that is, heating fuel. This finding means, for example,that a high efficiency window replacing similar low efficiency windowsin two different buildings in the same geographical location for twodifferent customer types, for instance, a residential customer versus anindustrial customer, has the same economic value, as long as the HVACsystem efficiencies and fuel prices are the same for these two differentcustomers.

This finding vastly simplifies the process of analyzing the value ofbuilding shell investments by fundamentally altering how the analysisneeds to be performed. Rather than requiring a full energy audit-styleanalysis of the building to assess any the costs and benefits of aparticular energy efficiency investment, only the investment ofinterest, the building's HVAC system efficiency, and the price and typeof fuel being saved are required.

As a result, the analysis of a building shell investment becomes muchmore like that of an appliance purchase, where the energy savings, forexample, equals the consumption of the old refrigerator minus the costof the new refrigerator, thereby avoiding the costs of a whole housebuilding analysis. Thus, a consumer can readily determine whether anacceptable return on investment will be realized in terms of costsversus likely energy savings. This result could be used in a variety ofplaces:

-   -   Direct display of economic impact in ecommerce sites. A Web        service that estimates economic value can be made available to        Web sites where consumers purchase building shell replacements.        The consumer would select the product they are purchasing, for        instance, a specific window, and would either specify the        product that they are replacing or a typical value can be        provided. This information would be submitted to the Web        service, which would then return an estimate of savings using        the input parameters described supra.    -   Tools for salespeople at retail and online establishments.    -   Tools for mobile or door-to-door sales people.    -   Tools to support energy auditors for immediate economic        assessment of audit findings. For example, a picture of a        specific portion of a house can be taken and the dollar value of        addressing problems can be attached.    -   Have a document with virtual sticky tabs that show economics of        exact value for each portion of the house. The document could be        used by energy auditors and other interested parties.    -   Available to companies interacting with new building purchasers        to interactively allow them to understand the effects of        different building choices from an economic (and environmental)        perspective using a computer program or Internet-based tool.    -   Enable real estate agents working with customers at the time of        a new home purchase to quantify the value of upgrades to the        building at the time of purchase.    -   Tools to simplify the optimization problem because most parts of        the problem are separable and simply require a rank ordering of        cost-benefit analysis of the various measures and do not require        detailed computer models that applied to specific houses.    -   The time to fix the insulation and ventilation in a homeowner's        attic is when during reroofing. This result could be integrated        into the roofing quoting tools.    -   Incorporated into a holistic zero net energy analysis computer        program or Web site to take an existing building to zero net        consumption.    -   Integration into tools for architects, builders, designers for        new construction or retrofit. Size building features or HVAC        system. More windows or less windows will affect HVAC system        size.

Application to Thermal Conductivity Analysis

A building's thermal conductivity can be characterized using onlymeasured utility billing data (natural gas and electricity consumption)and assumptions about effective window area, HVAC system efficiency andaverage indoor building temperature. This test could be used as follows:

-   -   Utilities lack direct methods to measure the energy savings        associated with building shell improvements. Use this test to        provide a method for electric utilities to validate energy        efficiency investments for their energy efficiency programs        without requiring an on-site visit or the typical detailed        energy audit. This method would help to address the measurement        and evaluation (M&E) issues currently associated with energy        efficiency programs.    -   HVAC companies could efficiently size HVAC systems based on        empirical results, rather than performing Manual J calculations        or using rules of thumb. This test could save customers money        because Manual J calculations require a detailed energy audit.        This test could also save customers capital costs since rules of        thumb typically oversize HVAC systems, particularly for        residential customers, by a significant margin.    -   A company could work with utilities (who have energy efficiency        goals) and real estate agents (who interact with customers when        the home is purchased) to identify and target inefficient homes        that could be upgraded at the time between sale and occupancy.        This approach greatly reduces the cost of the analysis, and the        unoccupied home offers an ideal time to perform upgrades without        any inconvenience to the homeowners.    -   Goals could be set for consumers to reduce a building's heating        needs to the point where a new HVAC system is avoided        altogether, thus saving the consumer a significant capital cost.

Application to Building Performance Studies

A building's performance can be fully characterized in terms of fourparameters using a suite of short-duration (several day) tests. The fourparameters include thermal conductivity, that is, heat losses, thermalmass, effective window area, and HVAC system efficiency. An assumptionis made about average indoor building temperature. These (or theprevious) characterizations could be used as follows:

-   -   Utilities could identify potential targets for building shell        investments using only utility billing data. Buildings could be        identified in a two-step process. First, thermal conductivity        can be calculated using only electric and natural gas billing        data, making the required assumptions presented supra. Buildings        that pass this screen could be the focus of a follow-up,        on-site, short-duration test.    -   The results from this test suite can be used to generate        detailed time series fuel consumption data (either natural gas        or electricity). This data can be combined with an economic        analysis tool, such as the PowerBill service        (http://www.cleanpower.com/products/powerbill/), a software        service offered by Clean Power Research, L.L.C., Napa, Calif.,        to calculate the economic impacts of the changes using detailed,        time-of-use rate structures.

Application to “Smart” Thermostat Users

The results from the short-duration tests, as described supra withreference to FIG. 4, could be combined with measured indoor buildingtemperature data collected using an Internet-accessible thermostat, suchas a Nest thermostat device or a Lyric thermostat device, cited supra,or other so-called “smart” thermostat devices, thereby avoiding havingto make assumptions about indoor building temperature. The buildingcharacterization parameters could then be combined with energyinvestment alternatives to educate consumers about the energy, economic,and environmental benefits associated with proposed purchases.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A method for estimating indoor temperature timeseries data of a building with the aid of a digital computer, comprisingthe steps of: finding thermal conductivity, thermal mass, effectivewindow area, and efficiency of an HVAC system of a building withinvolvement of a computer through empirical testing of the building overa monitored time frame; recording with involvement of the computer adifference between indoor and outdoor temperatures of the buildingduring the monitored time frame; defining with the computer a timeperiod for each interval in a time series; retrieving with the computeran average occupancy, internal electricity consumption, solar resource,rating of the HVAC system, and status of the HVAC system as applicableto the building over the monitored time frame; and building with thecomputer the time series comprising temperature data based on thebuilding's indoor temperature, thermal mass, thermal conductivity,temperature difference, occupancy, internal electricity consumption,effective window area, solar resource, and the rating, efficiency andstatus of the HVAC system, comprising: finding with the computer a datavalue representing an initial temperature at the beginning of the timeseries based on the building's indoor temperature, thermal mass, thermalconductivity, temperature difference, average occupancy, internalelectricity consumption, effective window area, solar resource, and therating, efficiency and status of the HVAC system; and recursivelygenerating each successive data value representing temperatures in theremainder of the time series by applying the temperature data valuemost-recently found for the time series, beginning with the initialtemperature data value, as the building's indoor temperature, wherein atleast one of heating of the building and cooling of the building isoptimized using the time series, the optimization comprising at leastone of an improvement to a shell of the building and changing a size ofthe HVAC system.
 2. A method according to claim 1, further comprisingthe step of: finding each successive data point T_(tΔt) ^(Indoor) in thetime series at time t+Δt in accordance with:$T_{t + {\Delta\; t}}^{Indoor} = {T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {\left( C_{1} \right)P_{t}} + {{Electric}_{t}\left( C_{2} \right)} + {W\;{{Solar}_{t}\left( C_{2} \right)}} + {{HeatOrCoolR}^{HVAC}\eta^{HVAC}{Status}_{t}}} \right\rbrack}\Delta\; t}}$where Δt represents the time period, T_(t) ^(Indoor) represents theindoor temperature at time t, M represents the thermal mass, UA^(Total)represents the thermal conductivity, T_(t) ^(Outdoor) represents theoutdoor temperature at time t, T_(t) ^(Indoor) represents the indoortemperature at time t, C₁ represents a conversion factor for occupancygains, P_(t) represents the average occupancy at time t, Electric_(t)represents internal electricity consumption at time t, C₂ represents apower unit conversion factor, W represents the effective window area,Solar_(t) represents the solar resource produced at time t, HeatOrCoolis a binary number that is 1 in the heating season and −1 in the coolingseason, R^(HVAC) represents the rating of the HVAC system, η^(HVAC)represents the efficiency of the HVAC system, and Status, represents thestatus of the HVAC system at time t.
 3. A method according to claim 1,further comprising the steps of: selecting a maximum indoor temperaturerecorded during the monitored time frame; establishing a limit for themaximum indoor temperature of the building as the minimum of the maximumrecorded indoor temperature and the temperature data value most-recentlyfound for the time series.
 4. A method according to claim 3, furthercomprising the step of: finding the limit for the maximum indoortemperature of the building T_(t+Δt) ^(Indoor) at time t in accordancewith:$T_{t + {\Delta\; t}}^{Indoor} = {{Min}\left\{ {T^{{Indoor} - {Max}},{T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {{Electric}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)} + {W\;{{Solar}_{t}\left( \frac{3\text{,}412\mspace{14mu}{Btu}}{1\mspace{14mu}{kWh}} \right)}} + {{HeatOrCoolR}^{HVAC}\eta^{HVAC}{Status}_{t}}} \right\rbrack}\Delta\; t}}} \right\}}$where T^(Indoor-Max) represents the maximum recorded indoor temperature,Δt represents the time period, T_(t) ^(Indoor) represents the indoortemperature at time t, M represents the thermal mass, UA^(Total)represents the thermal conductivity, T_(t) ^(Outdoor) represents theoutdoor temperature at time t, T_(t) ^(Indoor) represents the indoortemperature at time t, C₁ represents a conversion factor for occupancygains, P_(t) represents the average occupancy at time t, Electric_(t)represents internal electricity consumption at time t, C₂ represents apower unit conversion factor, W represents the effective window area,Solar_(t) represents the solar resource produced at time t, HeatOrCoolis a binary number that is 1 in the heating season and −1 in the coolingseason, R^(HVAC) represents the rating of the HVAC system, η^(HVAC)represents the efficiency of the HVAC system, and status_(t) representsthe status of the HVAC system at time t.
 5. A method according to claim1, further comprising the steps of: performing an empirical test over ashort duration during which the HVAC system is operated in the absenceof solar gain into the building, comprising the steps of: observing achange in the indoor temperature between the indoor temperature at thestart of the empirical test and the indoor temperature at the end of theempirical test; determining the indoor temperature as the average indoortemperature of the building over the short duration; determining theoutdoor temperature as the average outdoor temperature of the buildingover the short duration; determining the internal electricityconsumption as the average internal electricity consumption over theshort duration; basing the average occupancy as observed in the buildingduring the short duration; determining the solar resource as the averagesolar resource produced over the short duration; and determining theHVAC system status as observed in the building during the shortduration; and finding the HVAC system efficiency as a function of thethermal mass and the change in the indoor temperature, the thermalconductivity and the temperature difference, the average occupancy, andthe average internal electricity consumption, all over the rating andstatus of the HVAC system.
 6. A method according to claim 5, furthercomprising the steps of: finding the efficiency of the HVAC systemη^(HVAC) of the building in accordance with:$\eta^{HVAC} = {\left\lbrack {\frac{M\left( {T_{t + {\Delta\; t}}^{Indoor} - T_{t}^{Indoor}} \right)}{\Delta\; t} - {{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} - {\left( C_{1} \right)\overset{\_}{P}} - {\overset{\_}{Electric}\;\left( C_{2} \right)}} \right\rbrack\left\lbrack \frac{1}{{HeatOrCoolR}^{HVAC}\overset{\_}{Status}} \right\rbrack}$where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the short duration,T_(t+Δt) ^(Indoor) represents the ending indoor temperature, T_(t)^(Indoor) represents the starting indoor temperature, UA^(Total)represents the thermal conductivity, T ^(Indoor) represents the averageindoor temperature, T ^(Outdoor) represents the average outdoortemperature, C₁ represents a conversion factor for occupant heatinggains, P represents the average number of occupants, Electric representsaverage electricity consumption, C₂ represents a conversion factor fornon-HVAC electric heating gains, Status represents the HVAC systemstatus, HeatOrCool is a binary number that is 1 in the heating seasonand −1 in the cooling season, and R^(HVAC) represents the rating of theHVAC system.